MTH120 College Algebra Chapter 3.2

 3.2 Domain and Range

In mathematics, the domain and range are essential concepts when dealing with functions. They describe the set of possible input values (domain) and the set of possible output values (range) of a function. Let's define and explain each concept:

1. Domain: The domain of a function is the set of all possible input values (independent variable) for which the function is defined. It represents the values you can plug into the function to obtain a valid output. The domain is often expressed as a set of real numbers or a specific interval.

For example:

• For the function f(x) = √(x), the domain is all non-negative real numbers or [0, ∞).
• For the function g(x) = 1/(x - 2), the domain is all real numbers except x = 2, which is excluded because it would make the denominator zero.

2. Range: The range of a function is the set of all possible output values (dependent variable) that the function can produce based on the valid inputs from its domain. It represents the values the function can take on.

For example:

• For the function f(x) = x^2, the range is all non-negative real numbers or [0, ∞).
• For the function g(x) = sin(x), the range is between -1 and 1, inclusive, because the sine function oscillates between these values.

It's important to note that not all real numbers may be in the range of a given function, depending on the specific function and its behavior.

In some cases, you may need to analyze a function or its graph to determine its domain and range. For example, a rational function may have excluded values in its domain due to division by zero, and a trigonometric function may have a limited range due to its periodic behavior.

Understanding the domain and range of a function is crucial for various purposes, such as solving equations, finding the inverse of a function, and analyzing the behavior of functions in mathematical modeling and real-world applications.

To find the domain of a function defined by an equation, you need to determine the set of all valid input values (independent variable values) that make the function meaningful and well-defined. Here are some common steps to find the domain of a function:

1. Identify Any Restrictions: Examine the equation to identify any restrictions on the domain. These restrictions are typically related to operations that would result in undefined values, such as division by zero, square roots of negative numbers, or logarithms of non-positive numbers.

2. Analyze Square Roots and Radicals: If the equation contains square roots (√) or other radicals, consider what values can be taken under the radical. Square roots of negative numbers are undefined in the real number system.

3. Check for Fractional Expressions: If the equation contains fractions or rational expressions, check for values of the variable that would make the denominator equal to zero. These values should be excluded from the domain to avoid division by zero.

4. Look for Logarithmic and Exponential Functions: If the equation involves logarithmic or exponential functions, make sure the arguments of these functions are positive. Logarithms of non-positive numbers are undefined.

5. Consider Inequalities: If the equation involves inequalities, such as greater than (>) or less than (<) signs, consider the range of values that satisfy these inequalities.

6. Combine All Restrictions: Combine all the restrictions you've identified to determine the overall domain of the function. The domain should include all values that don't violate any of the identified restrictions.

7. Express the Domain: Express the domain using set notation or interval notation, depending on the form that best represents the set of valid input values.

Let's work through an example:

Example: Find the domain of the function defined by the equation:

$�(�)=\frac{1}{�-2}$

In this case:

• There is a restriction when the denominator $(�-2)$ becomes zero, so $�-2=0$. Solving for $�$, we find that $�=2$. Therefore, $�=2$ is not in the domain.

• There are no square roots, radicals, logarithms, or exponential functions in this equation.

• There are no inequalities to consider.

So, the domain of this function is all real numbers except $�=2$, expressed as:

$\text{Domain: }(-\mathrm{\infty },2)\cup (2,\mathrm{\infty })$

This means that the function is defined for all real numbers except $�=2$.

By following these steps and analyzing the equation for any restrictions, you can find the domain of a function defined by an equation.

The domain of a function is typically expressed as a set of input values (often denoted by "x") for which the function is defined. It's not common to express the domain as a set of ordered pairs, as ordered pairs are typically used to represent points on a graph rather than the domain itself. However, you can still determine the domain and then represent it as a set of ordered pairs if needed.

Let's work through an example and then represent its domain as a set of ordered pairs:

Example:

Consider the function $�(�)=\sqrt{�+3}$. To find the domain:

1. Identify any restrictions on the domain:

   In this case, the square root function is defined only for non-negative real numbers. Therefore, the expression inside the square root ($�+3$) must be greater than or equal to zero:

   $�+3\ge 0$

2. Solve for $�$:

   $�\ge -3$

This inequality tells us that the function $�(�)=\sqrt{�+3}$ is defined for all real numbers $�$ greater than or equal to -3.

Now, if you want to represent the domain as a set of ordered pairs, you can do so by pairing each valid $�$ value with the corresponding $�(�)$ value. However, the domain itself doesn't change; it's still the set of all real numbers greater than or equal to -3.

If you want to represent it as ordered pairs, you might do so like this:

$\text{Domain as Ordered Pairs: }\{(�,�(�))\mid �\ge -3\}$

In this notation, you are indicating that for any $�$ value greater than or equal to -3, there is a corresponding ordered pair $(�,�(�))$, where $�(�)=\sqrt{�+3}$. However, note that this notation is not commonly used for expressing the domain; it's more common to represent the domain simply as a set of $�$ values.

To find the domain of a function written in equation form that includes a fraction, you need to determine the values of the independent variable (usually denoted as "x") for which the function is defined. Specifically, you need to identify any values of x that would lead to division by zero or other operations that result in undefined values within the fraction. Here's how to find the domain step by step:

1. Identify the Denominator: Examine the fraction in the equation. The domain restrictions often arise from the denominator. Determine the expressions that appear in the denominator.

2. Set the Denominator Equal to Zero: For each expression in the denominator, set it equal to zero and solve for x. These are the values of x that would result in division by zero or other undefined operations.

3. Combine All Restrictions: Combine all the restrictions found in step 2 with any other restrictions that may exist in the equation, such as square roots of negative numbers or logarithms of non-positive numbers.

4. Express the Domain: Express the domain using set notation or interval notation, depending on what form best represents the set of valid input values.

Let's work through an example:

Example:

Find the domain of the function defined by the equation:

$�(�)=\frac{1}{{�}^2-4}$

In this case:

• The denominator ${�}^2-4$ is a quadratic expression, and it becomes zero when ${�}^2-4=0$.

• Solve for $�$:

  ${�}^2-4=0$ $(�-2)(�+2)=0$

  So, $�$ can be either 2 or -2.

These are the values of $�$ for which the function is undefined (division by zero). Therefore, the domain of this function is all real numbers except $�=2$ and $�=-2$, expressed as:

$\text{Domain: }(-\mathrm{\infty },-2)\cup (-2,2)\cup (2,\mathrm{\infty })$

This means that the function is defined for all real numbers except $�=2$ and $�=-2$.

Keep in mind that when dealing with fractions, division by zero is a common source of domain restrictions, but other operations within the equation may introduce additional restrictions based on the specific equation.

Let's work through a couple of examples to find the domain of functions involving a denominator:

Example 1:

Find the domain of the function defined by the equation:

$�(�)=\frac{1}{�-3}$

In this case:

1. Identify the denominator: The denominator is $�-3$.

2. Set the denominator equal to zero and solve for $�$:

   $�-3=0$ $�=3$

The value $�=3$ makes the denominator equal to zero. Therefore, it should be excluded from the domain.

3. Express the domain: The domain of this function is all real numbers except $�=3$, expressed as:

   $\text{Domain: }(-\mathrm{\infty },3)\cup (3,\mathrm{\infty })$

This means that the function is defined for all real numbers except $�=3$.

Example 2:

Find the domain of the function defined by the equation:

$�(�)=\frac{2�}{{�}^2-4}$

In this case:

1. Identify the denominator: The denominator is ${�}^2-4$.

2. Set the denominator equal to zero and solve for $�$:

   ${�}^2-4=0$ $(�-2)(�+2)=0$

   Solving for $�$:

   $�-2=0\Rightarrow �=2$ $�+2=0\Rightarrow �=-2$

The values $�=2$ and $�=-2$ make the denominator equal to zero. Therefore, they should be excluded from the domain.

3. Express the domain: The domain of this function is all real numbers except $�=2$ and $�=-2$, expressed as:

   $\text{Domain: }(-\mathrm{\infty },-2)\cup (-2,2)\cup (2,\mathrm{\infty })$

This means that the function is defined for all real numbers except $�=2$ and $�=-2$.

In both examples, we identified the denominator, found the values of $�$ that would make the denominator zero, and expressed the domain as a set of intervals. This process helps ensure that the function is defined for all other real numbers.

When dealing with functions that include even roots (such as square roots), it's important to consider the restrictions on the domain. The domain restrictions arise from the fact that even roots are defined only for non-negative values (or zero). Here are some steps to find the domain of a function with an even root:

1. Identify the Even Root: Identify the part of the function that involves an even root, such as $\sqrt{�}$ or $\sqrt[4]{�}$.

2. Set the Argument Inside the Root Greater than or Equal to Zero: The argument inside the even root must be greater than or equal to zero. Set the expression under the root sign greater than or equal to zero and solve for $�$.

3. Express the Domain: Express the domain using set notation or interval notation based on the solutions found in step 2.

Let's work through an example:

Example:

Find the domain of the function defined by the equation:

$�(�)=\sqrt{4-{�}^2}$

In this case:

1. Identify the even root: The function involves the square root, which is an even root.

2. Set the argument inside the square root greater than or equal to zero:

   $4-{�}^2\ge 0$

   Solve for $�$:

   ${�}^2\le 4$

   $-2\le �\le 2$

3. Express the domain: The domain of this function is all real numbers between -2 and 2, inclusive, expressed as:

   $\text{Domain: }[-2,2]$

This means that the function is defined for all real numbers between -2 and 2, inclusive.

In summary, when dealing with functions involving even roots, ensure that the argument inside the root is non-negative, and then express the domain accordingly. The process involves setting the argument greater than or equal to zero and solving for $�$.

Can there be functions in which the domain and range do not intersect at all?

Yes, there can be functions in which the domain and range do not intersect at all. Such functions are not surjective, meaning they do not cover the entire range or codomain.

Here are a few scenarios that illustrate this concept:

1. Restricted Domain: You can have a function that is defined only for a specific subset of the real numbers, and its range may be entirely different from that subset. For example, consider the function $�(�)={�}^2$ with the domain restricted to $�\ge 0$. In this case, the domain is the non-negative real numbers, but the range is the set of non-negative real numbers. The range doesn't include negative numbers.

2. Discontinuous Functions: Functions that are discontinuous at certain points can also have disjoint domains and ranges. For instance, consider the step function $�(�)=\{\begin{array}{ll}0& \text{if }�<0\\ 1& \text{if }�\ge 0\end{array}$. The domain is all real numbers, but the range is the set $\{0,1\}$, and these sets don't intersect.

3. Piecewise Functions: Piecewise functions often have different rules for different intervals, and this can lead to non-intersecting domains and ranges. For example, consider a piecewise function where $�(�)=�$ for $�\ge 0$ and $�(�)=2�$ for $�<0$. The domain consists of all real numbers, but the range doesn't include negative values.

In each of these cases, there is no intersection between the domain and range. The domain represents the set of valid input values, while the range represents the set of possible output values. In some functions, these sets may overlap partially or entirely, but in others, they may not intersect at all, depending on the nature of the function and its definition.

In mathematics, you can use various notations to specify the domain and range of a function. These notations help convey information about the valid input values (domain) and the possible output values (range) of the function. Here are some common notations for specifying domain and range:

Domain Notations:

1. Set Notation: You can specify the domain using set notation, where you list the valid input values within curly braces. For example:

   • $�=\{�\mid �\text{ is a real number and }�\ge 0\}$ represents the domain of non-negative real numbers.

2. Interval Notation: Interval notation is often used when the domain consists of a continuous range of values. For example:

   • $�=[�,�]$ represents the closed interval from $�$ to $�$, including both endpoints.
   • $�=(�,�)$ represents the open interval from $�$ to $�$, excluding both endpoints.

3. Inequalities: You can express domain restrictions using inequalities. For example:

   • $�=\{�\mid �>0\}$ represents the domain of positive real numbers.

Range Notations:

1. Set Notation: Specify the range using set notation, listing the possible output values within curly braces. For example:

   • $�=\{�\mid �\text{ is a real number and }�\ge 0\}$ represents the range of non-negative real numbers.

2. Interval Notation: Similar to the domain, interval notation can be used for the range when it consists of a continuous range of values. For example:

   • $�=[�,�]$ represents the closed interval from $�$ to $�$, including both endpoints.
   • $�=(�,�)$ represents the open interval from $�$ to $�$, excluding both endpoints.

3. Inequalities: Express range restrictions using inequalities. For example:

   • $�=\{�\mid �<5\}$ represents the range of real numbers less than 5.

When specifying the domain and range of a function, be sure to use the notation that best represents the characteristics of the sets of valid input and output values. Set notation is versatile and commonly used to provide a concise and clear description of domains and ranges, while interval notation is particularly useful for continuous intervals. Inequalities are helpful when you want to convey specific conditions for the domain or range.

Set-builder notation and interval notation are two common ways to represent sets of real numbers, intervals, or ranges in mathematics. Each notation has its own advantages and is used in different contexts.

Set-Builder Notation:

Set-builder notation is a concise and flexible way to describe a set of elements based on a rule or condition. It consists of the following components:

• The Set: The set itself is enclosed in curly braces {}. For example, $\{�\}$ represents a set of elements, where $�$ is the variable that satisfies a certain condition.

• The Variable: Inside the braces, you specify the variable that represents the elements of the set. This variable is often followed by a vertical bar or colon (| or :) to indicate that it is subject to a condition.

• The Condition: After the vertical bar or colon, you write the condition that the variable must satisfy to belong to the set. This condition can involve inequalities, equations, or any rule you want to impose.

• Ellipsis (optional): You can use ellipsis (...) to represent an infinite set or a set with a pattern. For example, $\{2�\mid �\in \mathbb{�}\}$ represents the set of even natural numbers.

Here are some examples of set-builder notation:

• The set of all natural numbers can be represented as $\{�\mid �\in \mathbb{�}\}$.
• The set of all real numbers greater than 2 can be represented as $\{�\mid �>2\}$.
• The set of all integers between -3 and 3 can be represented as $\{�\mid -3\le �\le 3\}$.
• The set of all prime numbers less than 10 can be represented as $\{�\mid �\text{ is prime and }2\le �<10\}$.

Interval Notation:

Interval notation is used to represent continuous sets or ranges of real numbers using brackets and parentheses. It is particularly useful when dealing with intervals along the real number line. There are several forms of interval notation:

• Closed Interval: Represented as $[�,�]$, where $�$ and $�$ are the endpoints of the interval, and both $�$ and $�$ are included in the set. It includes all numbers between $�$ and $�$, including $�$ and $�$.

• Open Interval: Represented as $(�,�)$, where $�$ and $�$ are the endpoints of the interval, but neither $�$ nor $�$ is included in the set. It includes all numbers strictly between $�$ and $�$.

• Half-Open or Half-Closed Interval: Represented as $[�,�)$ or $(�,�]$, where one endpoint is included (closed) and the other is not (open).

• Unbounded Interval: Represented as $(-\mathrm{\infty },�)$, $(�,\mathrm{\infty })$, or $(-\mathrm{\infty },\mathrm{\infty })$, where the interval extends indefinitely in one or both directions.

Here are some examples of interval notation:

• The closed interval from 1 to 5 is represented as $[1,5]$.
• The open interval from -2 to 2 is represented as $(-2,2)$.
• The half-open interval from 0 to 3 is represented as $[0,3)$.
• The unbounded interval to the left of 4 is represented as $(-\mathrm{\infty },4)$.
• The set of all real numbers is represented as $(-\mathrm{\infty },\mathrm{\infty })$.

Both set-builder notation and interval notation have their advantages, and you can choose the notation that best suits your needs and the context of the mathematical problem you are working on.

When given a line graph, you can determine the set of values (interval) represented by the graph and express it using interval notation. Interval notation is particularly useful for representing continuous sets or ranges along the real number line. Here are some examples:

Example 1: Closed Interval

Consider a line graph that represents a solid line segment from $�=-2$ to $�=3$. This means that all the values between -2 and 3, including -2 and 3, are part of the set. In interval notation, this can be represented as:

$[-2,3]$

This notation indicates that the set includes all real numbers greater than or equal to -2 and less than or equal to 3.

Example 2: Open Interval

Now, imagine a line graph with an open circle at $�=1$ and another open circle at $�=4$, indicating that the values at these points are not included in the set. The line segment between these points is part of the set. In interval notation, this can be represented as:

$(1,4)$

This notation indicates that the set includes all real numbers greater than 1 and less than 4.

Example 3: Half-Open or Half-Closed Intervals

Suppose you have a line graph with a closed circle at $�=-3$ and an open circle at $�=2$. This means that the set includes -3 but not 2. In interval notation, you can represent this as a half-open or half-closed interval:

$[-3,2)$

This notation indicates that the set includes all real numbers greater than or equal to -3 and less than 2.

Example 4: Unbounded Intervals

In some cases, a line graph may extend infinitely in one or both directions. For example, if you have a horizontal line that covers all real numbers greater than or equal to 0, you can represent this as an unbounded interval:

$[0,\mathrm{\infty })$

This notation indicates that the set includes all real numbers greater than or equal to 0 and extends infinitely to the right.

In summary, when given a line graph, carefully analyze the endpoints (open or closed circles) and the direction of the line to determine the set of values it represents. Then, use the appropriate interval notation to express the set.

To find the domain and range of a function from its graph, you can visually examine the graph and determine the set of valid input values (domain) and the set of possible output values (range). Here's how to do it:

Finding the Domain:

The domain of a function represents all the possible input values (x-values) for which the function is defined. To find the domain from the graph:

1. Examine the x-values: Look at the horizontal axis (x-axis) of the graph. Identify the range of x-values for which the graph is drawn. This will give you the domain.

2. Consider any restrictions: Pay attention to any vertical asymptotes, holes, or discontinuities in the graph. These points might indicate values that are excluded from the domain due to division by zero or other undefined operations.

3. Express the domain: Write down the domain as a set of valid x-values or using interval notation if it represents a continuous interval.

Finding the Range:

The range of a function represents all the possible output values (y-values) that the function can produce. To find the range from the graph:

1. Examine the y-values: Look at the vertical axis (y-axis) of the graph. Identify the range of y-values that correspond to the points on the graph.

2. Consider the highest and lowest points: Determine the highest and lowest points on the graph. These will be the maximum and minimum values in the range, respectively.

3. Express the range: Write down the range as a set of valid y-values or using interval notation if it represents a continuous interval. If the graph extends infinitely in either direction, you can use $-\mathrm{\infty }$ or $\mathrm{\infty }$ in the range notation.

Let's look at an example:

Example:

Consider the graph of the function $�(�)=\frac{1}{�}$:

• Domain: From the graph, you can see that the function is defined for all real numbers except $�=0$, as there is a vertical asymptote at $�=0$. So, the domain is $�\in \mathbb{�}$ but $�\mathrm{\ne }0$, which can be expressed as $\{�\mid �\in \mathbb{�},�\mathrm{\ne }0\}$.

• Range: The range of the function includes all real numbers except $�=0$. You can see that the function approaches $\mathrm{\infty }$ as $�$ approaches 0 from the right and approaches $-\mathrm{\infty }$ as $�$ approaches 0 from the left. So, the range is $�\in \mathbb{�}$ but $�\mathrm{\ne }0$, which can be expressed as $\{�\mid �\in \mathbb{�},�\mathrm{\ne }0\}$.

By visually examining the graph and considering any restrictions or asymptotes, you can determine the domain and range of the function.

To find the domain and range from a graph of oil production, you would examine the graph and identify the relevant information about the production levels (range) and the time periods (domain). Here's how you can do it using an example:

Example: Oil Production Over Time

Suppose you have a graph that represents oil production over time, with time measured in years and oil production measured in barrels per year.

1. Finding the Domain (Time Period):

   • Examine the x-axis: Look at the horizontal axis (x-axis) of the graph, which represents time in years. Identify the range of years for which the oil production data is provided.

   • Consider any gaps or interruptions: Check if there are any gaps or interruptions in the time period for which oil production data is available. If there are, they should be noted as excluded from the domain.

   • Express the domain: Write down the domain as a range of years, for example, "Domain: [1990, 2020]" if the graph covers the years 1990 to 2020.

2. Finding the Range (Oil Production Levels):

   • Examine the y-axis: Look at the vertical axis (y-axis) of the graph, which represents oil production measured in barrels per year. Identify the range of production levels shown on the graph.

   • Consider the highest and lowest points: Determine the maximum and minimum oil production levels represented on the graph. These will be the upper and lower bounds of the range.

   • Express the range: Write down the range as a set of oil production levels, for example, "Range: [0 barrels/year, 10,000 barrels/year]" if the graph shows oil production levels ranging from 0 to 10,000 barrels per year.

Remember that the domain represents the time period for which oil production data is available, and the range represents the possible oil production levels during that time period based on the graph. The graph may provide additional insights, such as trends or fluctuations in oil production over time.

In practice, the domain and range may be represented as intervals or sets of values, depending on the specifics of the graph and the data it conveys.

The domain and range of "toolkit" functions are well-defined and typically follow certain patterns. Toolkit functions are a set of common and frequently used functions in mathematics that serve as building blocks for more complex functions. Here are the domains and ranges of some common toolkit functions:

1. Linear Function (y = mx + b):

• Domain: The domain is all real numbers ($\mathbb{�}$). Linear functions are defined for all real values of $�$.

• Range: The range is also all real numbers ($\mathbb{�}$). A linear function can produce any real number as its output.

2. Quadratic Function (y = ax^2 + bx + c):

• Domain: The domain is all real numbers ($\mathbb{�}$). Quadratic functions are defined for all real values of $�$.

• Range: The range depends on the sign of the coefficient $�$. If $�>0$, the range is $[�,\mathrm{\infty })$ where $�$ is the minimum point on the parabola. If $�<0$, the range is $(-\mathrm{\infty },�]$.

3. Absolute Value Function (y = |x|):

• Domain: The domain is all real numbers ($\mathbb{�}$). The absolute value function is defined for all real values of $�$.

• Range: The range is $[0,\mathrm{\infty })$. The absolute value of any real number is non-negative.

4. Square Root Function (y = √x):

• Domain: The domain is $[0,\mathrm{\infty })$. The square root function is defined for non-negative real values of $�$.

• Range: The range is $[0,\mathrm{\infty })$. The square root of a non-negative number is non-negative.

5. Exponential Function (y = a^x):

• Domain: The domain is all real numbers ($\mathbb{�}$). Exponential functions are defined for all real values of $�$.

• Range: The range is $(0,\mathrm{\infty })$ if $�>1$ and $(0,1]$ if $0<�<1$. Exponential functions with positive bases never equal zero and are always positive (or between 0 and 1).

6. Logarithmic Function (y = log_a(x)):

• Domain: The domain is $(0,\mathrm{\infty })$ for a logarithmic function. The argument of the logarithm must be a positive real number.

• Range: The range is all real numbers ($\mathbb{�}$). Logarithmic functions can output any real number.

These are general properties of toolkit functions. However, keep in mind that specific functions may have additional restrictions or variations based on the parameters and transformations applied to them. Always consider the specific form of the function and any additional conditions when determining the domain and range.

Let's go through examples of finding the domain and range using toolkit functions:

Example 1: Linear Function

Consider the linear function $�(�)=3�-2$.

• Domain: Linear functions have a domain of all real numbers ($\mathbb{�}$). So, the domain of this function is $\mathbb{�}$.

• Range: Linear functions also have a range of all real numbers ($\mathbb{�}$). So, the range of this function is $\mathbb{�}$.

Example 2: Quadratic Function

Let's take the quadratic function $�(�)={�}^2-4�+3$.

• Domain: Quadratic functions have a domain of all real numbers ($\mathbb{�}$). So, the domain of this function is $\mathbb{�}$.

• Range: To find the range, we need to determine the vertex of the parabola. The vertex formula is $�=-\frac{�}{2�}$, where $�$ and $�$ are the coefficients of ${�}^2$ and $�$ in the quadratic equation. In this case, $�=1$ and $�=-4$.

  Using the formula, we find $�=-\frac{-4}{2(1)}=2$. So, the vertex occurs at $�=2$.

  Now, plug $�=2$ into the function to find the corresponding $�$-coordinate: $�(2)=2^2-4(2)+3=4-8+3=-1$.

  The vertex is $(2,-1)$, and since this is a parabola opening upward, the range is all real numbers greater than or equal to -1. Therefore, the range is $[-1,\mathrm{\infty })$.

Example 3: Absolute Value Function

Consider the absolute value function $ℎ(�)=\mathrm{\mid }�-3\mathrm{\mid }$.

• Domain: Absolute value functions have a domain of all real numbers ($\mathbb{�}$). So, the domain of this function is $\mathbb{�}$.

• Range: The range of this absolute value function is $[0,\mathrm{\infty })$. Absolute value functions always output non-negative values.

Example 4: Square Root Function

Let's look at the square root function $�(�)=\sqrt{�+2}$.

• Domain: Square root functions have a domain of all real numbers $[0,\mathrm{\infty })$. So, the domain of this function is $[0,\mathrm{\infty })$.

• Range: The range of this square root function is $[0,\mathrm{\infty })$. Square root functions always output non-negative values.

These examples illustrate how to determine the domain and range of functions using common toolkit functions. It's essential to understand the properties and behavior of each function type to correctly identify the domain and range.

Graphing piecewise-defined functions involves plotting multiple functions or expressions on the same coordinate system, each of which is valid for a specific interval or range of input values. Here's a step-by-step guide on how to graph a piecewise-defined function:

Step 1: Identify the Function Pieces

Start by identifying the different pieces of the piecewise function and the intervals for which each piece is valid. Each piece of the function corresponds to a specific condition or range of input values.

Step 2: Plot Each Piece Separately

For each piece of the function, plot the corresponding graph separately within its valid interval. Follow these guidelines:

• Use the equation or expression associated with each piece to determine the y-values (output values) for various x-values within the interval.

• Pay attention to any transformations (shifts, stretches, or reflections) applied to each piece. These transformations may affect the shape and position of the graph.

• If there are vertical asymptotes or holes in the graph due to discontinuities, make sure to account for them.

Step 3: Combine the Graphs

Combine the individual graphs of the function pieces on the same coordinate system. Ensure that each graph is correctly positioned according to its valid interval.

• If a piece is valid for an open interval (e.g., $0<�<5$), use an open circle or an open endpoint to indicate that the graph does not include those specific values.

• If a piece is valid for a closed interval (e.g., $-2\le �\le 2$), use a solid line or closed endpoint to indicate that the graph includes those specific values.

Step 4: Label and Annotate

Label the axes with appropriate variable names (usually x and y). Provide axis scales and labels if necessary. Label the graphs of each piece with their respective equations or expressions.

Step 5: Test Points

Test some points within each interval to ensure that the graph correctly represents the function's behavior. This can help you verify the accuracy of your graph.

Step 6: Highlight Discontinuities

If there are discontinuities (jumps, holes, or asymptotes) in the graph, highlight them and explain the reason for each discontinuity, such as division by zero, piecewise conditions, or vertical asymptotes.

Step 7: Provide the Domain

Specify the domain of the piecewise function, which is the set of all x-values for which the function is defined. The domain should be based on the union of the valid intervals for each piece.

Graphing piecewise functions can be more complex than graphing single functions, especially if there are multiple pieces with different behaviors. It's crucial to pay attention to the specific conditions and intervals for each piece and accurately represent the function's behavior within those intervals.

Let's break down the notation used in the piecewise function representation:

1. ${�}_1(�),{�}_2(�),\dots ,{�}_{�}(�)$: These are different functions or expressions. Each one represents a mathematical formula or rule that applies to a specific interval or condition within the domain of the piecewise function.

2. $�,�,�,\dots ,�$: These are real numbers that define the boundaries or conditions for each piece of the function. These numbers represent the values of the independent variable $�$ where the function transitions from one rule to another.

3. $\le$, $<$, $\ge$: These symbols are used to specify the range of values for which each rule or expression is valid. Here's what each of them means:

   • $\le$: Less than or equal to (inclusive). For example, $�\le �$ means $�$ can take on the value of $�$ or any value greater than $�$.
   • $<$: Less than (exclusive). For example, $�<�$ means $�$ can take any value greater than $�$, but not equal to $�$.
   • $\ge$: Greater than or equal to (inclusive). For example, $�\ge �$ means $�$ can take on the value of $�$ or any value greater than $�$.

Now, let's interpret the piecewise function using the provided notation:

• ${�}_1(�)$ is the first function, and it applies when $�\le �<�$. This means that the rule represented by ${�}_1(�)$ is valid for values of $�$ that are greater than or equal to $�$ but less than $�$.

• ${�}_2(�)$ is the second function, and it applies when $�\le �<�$. This means that the rule represented by ${�}_2(�)$ is valid for values of $�$ that are greater than or equal to $�$ but less than $�$.

• ${�}_{�}(�)$ is the last function, and it applies when $�\ge �$. This means that the rule represented by ${�}_{�}(�)$ is valid for values of $�$ that are greater than or equal to $�$.

Each of these functions, ${�}_1(�),{�}_2(�),\dots ,{�}_{�}(�)$, applies to a specific range of values of $�$ as determined by the inequalities $�\le �<�$, $�\le �<�$, and $�\ge �$. The piecewise function allows you to define different rules for different parts of the domain, which can be useful in various mathematical and real-world applications.

To graph the piecewise function $�(�)$, we'll plot the different cases separately for the given intervals. Here's how you can graph it:

1. For $�<-1$: In this interval, $�(�)={�}^3-2�-\sqrt{-�}$. We will plot this function for values of $�$ less than -1.

2. For $-1<�<4$: In this interval, $�(�)=-1$. This is a constant value, so we'll plot a horizontal line at $�=-1$ for values of $�$ between -1 and 4.

3. For $�>4$: In this interval, $�(�)=4�$. We will plot a linear function with a slope of 4 for values of $�$ greater than 4.

Let's create a graph for each interval separately and then combine them:

For $�<-1$: We'll plot the function $�(�)={�}^3-2�-\sqrt{-�}$ for $�$ in this range. However, note that $\sqrt{-�}$ is only defined for $�\le 0$ since the square root of a negative number is not real. So, the function simplifies to $�(�)={�}^3-2�$ for $�<-1$.

For $-1<�<4$: $�(�)=-1$ is a horizontal line at $�=-1$ for $�$ in this range.

For $�>4$: $�(�)=4�$ is a linear function with a slope of 4 for $�$ greater than 4.

Now, let's combine these graphs:

• For $�<-1$, we'll plot $�(�)={�}^3-2�$ as a cubic curve.
• For $-1<�<4$, we'll have a horizontal line at $�=-1$.
• For $�>4$, we'll have a linear function $�(�)=4�$ with a slope of 4.

The resulting graph will be a combination of these three components, each corresponding to its respective interval.