MTH120 College Algebra Chapter 3.7

 3.7 Inverse Functions:

Inverse functions are a fundamental concept in mathematics that are used to reverse the effects of a function. An inverse function "undoes" the operations of the original function, allowing you to retrieve the original input value from the output value. In this explanation, I'll provide an overview of inverse functions and how they work:

1. Definition of an Inverse Function:

   • Given a function f(x), its inverse, denoted as f^(-1)(x), is another function such that when you apply f and then f^(-1), or vice versa, you get back the original value:

     f^(-1)(f(x)) = x for all x in the domain of f, and f(f^(-1)(x)) = x for all x in the domain of f^(-1).

   In other words, applying the function and its inverse in either order results in the identity function.

2. Finding the Inverse Function:

   • To find the inverse of a function, you typically follow these steps: a. Replace f(x) with y. b. Swap the roles of x and y, making the equation y = f(x). c. Solve for y to express y as a function of x. d. Replace y with f^(-1)(x) to get the inverse function.

   For example, if you have the function f(x) = 2x + 3, you would: a. Replace f(x) with y: y = 2x + 3. b. Swap x and y: x = 2y + 3. c. Solve for y: y = (x - 3)/2. d. Replace y with f^(-1)(x): f^(-1)(x) = (x - 3)/2.

3. Graphical Interpretation:

   • On a graph, if you have a function f(x), the graph of its inverse function f^(-1)(x) is a reflection of the original graph across the line y = x. This means that the points (x, y) on the graph of f(x) become (y, x) on the graph of f^(-1)(x).

4. Important Note:

   • Not all functions have inverses. For an inverse function to exist, the original function must be one-to-one (injective), meaning that each input value corresponds to a unique output value. In other words, the function must pass the horizontal line test, where no horizontal line intersects the graph more than once.

5. Examples:

   • For the function f(x) = 2x + 3, its inverse is f^(-1)(x) = (x - 3)/2.
   • For the function f(x) = e^x (the exponential function), its inverse is the natural logarithm function, f^(-1)(x) = ln(x).

Inverse functions are widely used in mathematics, engineering, physics, and other fields to solve equations, model real-world problems, and perform transformations. They are a crucial concept for understanding the relationships between functions and their reversibility.

Verifying that two functions are inverse functions involves checking whether they satisfy the definition of inverse functions, which means that when you apply one function to the output of the other, you get back the original input. Here are the steps to verify that two functions are inverses of each other:

1. Given Functions: You have two functions, f(x) and g(x), and you want to verify if they are inverses of each other.

2. Compose the Functions: Compute the composition of the two functions in both orders:

   • First, find the composition of f(g(x)), which means you apply g(x) to the output of f(x).
   • Next, find the composition of g(f(x)), which means you apply f(x) to the output of g(x).

3. Check for Identity: Verify that both compositions result in the identity function, which is typically the function that returns the input value unchanged. For example, if you're working with real numbers, the identity function is often represented as I(x) = x.

   • If f(g(x)) = I(x) and g(f(x)) = I(x) for all x in their domains, then f(x) and g(x) are inverse functions of each other.

Here's a step-by-step example to illustrate this process:

Example: Verify if the functions f(x) = 2x + 3 and g(x) = (x - 3) / 2 are inverses of each other.

1. Given Functions: You have f(x) = 2x + 3 and g(x) = (x - 3) / 2.

2. Compose the Functions:

   • First, compute f(g(x)): f(g(x)) = f((x - 3) / 2) = 2((x - 3) / 2) + 3 (applying f(x) to g(x)) = (x - 3) + 3 (simplifying) = x

   • Next, compute g(f(x)): g(f(x)) = g(2x + 3) = ((2x + 3) - 3) / 2 (applying g(x) to f(x)) = (2x) / 2 = x

3. Check for Identity: Both compositions result in the identity function I(x) = x for all x in their domains.

Since f(g(x)) = x and g(f(x)) = x for all x, you can conclude that f(x) = 2x + 3 and g(x) = (x - 3) / 2 are indeed inverse functions of each other.

This verification process confirms that the two functions undo each other's operations and satisfy the definition of inverse functions.

An inverse function is a concept in mathematics that refers to a function that "undoes" the action of another function. In other words, if you have a function f(x) that takes an input x and produces an output f(x), its inverse, denoted as f^(-1)(x), takes the output f(x) and returns the original input x. In simple terms, an inverse function reverses the process of the original function.

Here are key points to understand about inverse functions:

1. Existence of Inverse Functions:

   • Not all functions have inverse functions. For an inverse function to exist, the original function must be a one-to-one (injective) function. This means that each input value corresponds to a unique output value, and no two different input values produce the same output value. In graphical terms, the function must pass the horizontal line test, which ensures that no horizontal line intersects the graph of the function more than once.

2. Definition:

   • Given a function f(x), its inverse function, denoted as f^(-1)(x), satisfies the following:
     • f^(-1)(f(x)) = x for all x in the domain of f.
     • f(f^(-1)(x)) = x for all x in the domain of f^(-1).

3. Notation:

   • The notation for an inverse function is f^(-1)(x), where the "^(-1)" indicates the inverse. It is not an exponent.

4. Graphical Interpretation:

   • On a graph, the graph of an inverse function is a reflection of the original function across the line y = x. This means that the points (x, y) on the graph of f(x) become (y, x) on the graph of f^(-1)(x). This reflection property is a key characteristic of inverse functions.

5. Finding the Inverse Function:

   • To find the inverse function, you typically follow these steps:
     • Replace f(x) with y.
     • Swap the roles of x and y, making the equation y = f(x).
     • Solve for y to express y as a function of x.
     • Replace y with f^(-1)(x) to get the inverse function.

Inverse functions play a crucial role in mathematics and have applications in various fields, including solving equations, cryptography, optimization problems, and more. They are used to reverse the effects of functions and recover original data or values from transformed data.

Here are some examples of functions and their corresponding inverse functions:

1. Linear Function and Its Inverse:

Function: $�(�)=2�+3$

To find the inverse function:

1. Replace $�(�)$ with $�$: $�=2�+3$.
2. Swap the roles of $�$ and $�$: $�=2�+3$.
3. Solve for $�$: $�=\frac{�-3}{2}$.

So, the inverse function is ${�}^{-1}(�)=\frac{�-3}{2}$.

2. Exponential Function and Its Inverse:

Function: $�(�)={�}^{�}$ (the exponential function)

Inverse Function: ${�}^{-1}(�)=\ln (�)$ (the natural logarithm)

These two functions are inverses of each other because applying one function to the output of the other gives you back the original input.

3. Quadratic Function and Its Inverse:

Function: $�(�)={�}^2$ (a simple quadratic function)

This function does not have an inverse because it is not one-to-one. For example, both $�=2$ and $�=-2$ would map to the same value $�(�)=4$, violating the one-to-one condition required for an inverse.

4. Trigonometric Functions and Their Inverses:

Function: $�(�)=\sin (�)$ (the sine function)

Inverse Function: ${�}^{-1}(�)=\arcsin (�)$ (the arcsines or inverse sine function)

Function: $�(�)=\cos (�)$ (the cosine function)

Inverse Function: ${�}^{-1}(�)=\arccos (�)$ (the arccosines or inverse cosine function)

These pairs of functions are inverses of each other because they undo each other's effects. For example, $\sin (\arcsin (�))=�$ and $\cos (\arccos (�))=�$ for all $�$ within their domains.

These examples illustrate various types of functions and their inverse functions, highlighting the concept that an inverse function "undoes" the action of the original function.

To find the domain and range of an inverse function, you typically need to understand the relationship between the original function and its inverse. Here are the steps to determine the domain and range of an inverse function:

Domain of the Inverse Function:

1. Start with the original function $�(�)$ and consider its domain. The domain of $�(�)$ consists of all the values of $�$ for which $�(�)$ is defined.

2. Determine the range of the original function $�(�)$. The range of $�(�)$ consists of all the values that $�(�)$ can take as it varies over its domain.

3. The domain of the inverse function ${�}^{-1}(�)$ will be equal to the range of the original function $�(�)$.

Range of the Inverse Function:

1. Start with the original function $�(�)$ and consider its range. The range of $�(�)$ consists of all the values that $�(�)$ can take as it varies over its domain.

2. Determine the domain of the original function $�(�)$. The domain of $�(�)$ consists of all the values of $�$ for which $�(�)$ is defined.

3. The range of the inverse function ${�}^{-1}(�)$ will be equal to the domain of the original function $�(�)$.

It's important to remember that the domain and range of a function are closely related to its behavior and the specific values for which it is defined. When finding the domain and range of an inverse function, you are essentially interchanging the roles of $�$ and $�$ and considering the behavior of the original function in reverse.

Here's a simplified example to illustrate the process:

Example:

Consider the function $�(�)=2�+3$.

1. Domain of $�(�)$: The domain consists of all real numbers because there are no restrictions on $�$.

2. Range of $�(�)$: The range consists of all real numbers because $�(�)$ can take any real value as it varies over its domain.

3. Domain of ${�}^{-1}(�)$: The domain of the inverse function will be equal to the range of $�(�)$, which is all real numbers.

4. Range of ${�}^{-1}(�)$: The range of the inverse function will be equal to the domain of $�(�)$, which is all real numbers.

So, for this example, both the domain and range of the inverse function ${�}^{-1}(�)$ are all real numbers.

In more complex cases or with more specific functions, the domain and range of the inverse function may be subject to additional restrictions based on the properties of the original function.

To determine the domain and range of an inverse function, you need to understand the relationship between the original function and its inverse. Here are the general steps to find the domain and range of an inverse function:

Domain of the Inverse Function:

1. Start with the original function $�(�)$ and consider its domain. The domain of $�(�)$ consists of all the values of $�$ for which $�(�)$ is defined.

2. Determine the range of the original function $�(�)$. The range of $�(�)$ consists of all the values that $�(�)$ can take as it varies over its domain.

3. The domain of the inverse function ${�}^{-1}(�)$ will be equal to the range of the original function $�(�)$.

Range of the Inverse Function:

1. Start with the original function $�(�)$ and consider its range. The range of $�(�)$ consists of all the values that $�(�)$ can take as it varies over its domain.

2. Determine the domain of the original function $�(�)$. The domain of $�(�)$ consists of all the values of $�$ for which $�(�)$ is defined.

3. The range of the inverse function ${�}^{-1}(�)$ will be equal to the domain of the original function $�(�)$.

Now, let's look at some specific examples:

Example 1:

Consider the function $�(�)=2�+3$.

1. Domain of $�(�)$: The domain consists of all real numbers because there are no restrictions on $�$.

2. Range of $�(�)$: The range consists of all real numbers because $�(�)$ can take any real value as it varies over its domain.

3. Domain of ${�}^{-1}(�)$: The domain of the inverse function will be equal to the range of $�(�)$, which is all real numbers.

4. Range of ${�}^{-1}(�)$: The range of the inverse function will be equal to the domain of $�(�)$, which is all real numbers.

So, for this example, both the domain and range of the inverse function ${�}^{-1}(�)$ are all real numbers.

Example 2:

Consider the function $�(�)=\sqrt{�}$.

1. Domain of $�(�)$: The domain consists of non-negative real numbers ($�\ge 0$) because the square root is only defined for non-negative values of $�$.

2. Range of $�(�)$: The range consists of non-negative real numbers ($�\ge 0$) because the square root always produces non-negative results.

3. Domain of ${�}^{-1}(�)$: The domain of the inverse function will be equal to the range of $�(�)$, which is non-negative real numbers.

4. Range of ${�}^{-1}(�)$: The range of the inverse function will be equal to the domain of $�(�)$, which is non-negative real numbers.

In summary, the domain and range of an inverse function are determined by interchanging the roles of $�$ and $�$ and considering the behavior of the original function in reverse. The specific domain and range may vary depending on the original function and any restrictions it imposes.

Finding and evaluating inverse functions involves several steps, and it's essential to understand the relationship between the original function and its inverse. Here's a step-by-step guide:

Finding the Inverse Function:

1. Start with the original function, $�(�)$, and express it as $�=�(�)$.

2. Swap the roles of $�$ and $�$, which means you replace $�$ with $�$ and $�$ with $�$. This gives you an equation in the form $�=�(�)$.

3. Solve the equation for $�$ to express $�$ as a function of $�$. This will be your inverse function, denoted as ${�}^{-1}(�)$.

4. The inverse function ${�}^{-1}(�)$ should be written as $�={�}^{-1}(�)$.

Evaluating the Inverse Function:

Once you have found the inverse function ${�}^{-1}(�)$, you can use it to evaluate ${�}^{-1}(�)$ for specific values of $�$. Here's how:

1. Start with the inverse function $�={�}^{-1}(�)$.

2. Choose a specific value for $�$, let's call it ${�}_0$.

3. Substitute ${�}_0$ for $�$ in the inverse function to find the corresponding $�$ value. This will give you ${�}_0={�}^{-1}({�}_0)$.

4. The value ${�}_0$ is the result of evaluating the inverse function ${�}^{-1}(�)$ at $�={�}_0$. In other words, it's the image of ${�}_0$ under the inverse function.

Here's a specific example to illustrate these steps:

Example:

Consider the original function $�(�)=2�+3$. Let's find and evaluate its inverse function.

Finding the Inverse Function:

1. Start with the original function $�(�)$: $�=2�+3$.

2. Swap the roles of $�$ and $�$: $�=2�+3$.

3. Solve for $�$: $�-3=2�$ $�=\frac{�-3}{2}$

So, the inverse function is ${�}^{-1}(�)=\frac{�-3}{2}$.

Evaluating the Inverse Function:

Let's evaluate ${�}^{-1}(�)$ for a specific value of $�$, say ${�}_0=7$.

1. Start with the inverse function: $�=\frac{�-3}{2}$.

2. Substitute ${�}_0=7$ for $�$: $�=\frac{7-3}{2}=\frac{4}{2}=2$.

So, when $�=7$, the value of the inverse function ${�}^{-1}(�)$ is $�=2$.

You can follow these steps to find and evaluate the inverse function for any given original function and specific input values.

Evaluating the inverse of a function given a graph of the original function can be done graphically. Here are the steps to evaluate the inverse function using the graph of the original function:

1. Understand the Graph: Start by examining the graph of the original function. Look for points and understand the behavior of the function.

2. Identify a Point: Choose a specific point on the graph of the original function. Let's call this point $(�,�)$, where $�$ is the x-coordinate and $�$ is the y-coordinate.

3. Find the Inverse: To evaluate the inverse function at a specific point, swap the x and y coordinates of the chosen point. In other words, if the original function has a point $(�,�)$, the inverse function will have a corresponding point $(�,�)$.

4. Interpret the Result: The point $(�,�)$ is a point on the graph of the inverse function. It represents the value of the inverse function evaluated at $�=�$, which is ${�}^{-1}(�)=�$.

5. Repeat as Needed: You can repeat this process for other points on the graph of the original function to evaluate the inverse function at different values of $�$.

6. Domain and Range: Keep in mind that the domain and range of the inverse function can be determined by examining the behavior of the original function on the graph.

Here's a simple example:

Example:

Consider the original function $�(�)=2�+1$ and its graph. You want to evaluate the inverse function ${�}^{-1}(�)$ at a specific value.

1. Examine the graph of $�(�)$ and understand its behavior.

2. Choose a point on the graph, such as $(3,7)$, where $�=3$ and $�=7$.

3. Swap the x and y coordinates to find a point on the graph of ${�}^{-1}(�)$: $(7,3)$.

4. Interpret the result: ${�}^{-1}(7)=3$. This means that when $�=7$, the value of the inverse function is ${�}^{-1}(7)=3$.

5. You can repeat this process for other points on the graph to evaluate the inverse function at different values of $�$.

By examining the graph and swapping the coordinates, you can evaluate the inverse function at specific values. Remember that the domain and range of the inverse function can be inferred from the graph of the original function.

Finding the inverse of a function represented by a formula involves a series of algebraic steps. Here are the general steps to find the inverse of a function:

Step 1: Start with the original function, represented as $�=�(�)$.

Step 2: Replace $�(�)$ with $�$:

• Replace $�(�)$ with $�$ to get an equation in the form $�=\dots$.

Step 3: Swap the roles of $�$ and $�$:

• Replace $�$ with $�$ and $�$ with $�$ in the equation obtained from Step 2.

Step 4: Solve for $�$:

• Solve the equation from Step 3 for $�$ to express $�$ as a function of $�$. This will be your inverse function.

Step 5: Replace $�$ with ${�}^{-1}(�)$:

• Replace $�$ with ${�}^{-1}(�)$ to write the inverse function in the form ${�}^{-1}(�)=\dots$.

Here's a detailed example to illustrate these steps:

Example: Find the inverse of the function $�(�)=2�+3$.

Step 1: Start with the original function, $�=2�+3$.

Step 2: Replace $�(�)$ with $�$: $�=2�+3$

Step 3: Swap the roles of $�$ and $�$: $�=2�+3$

Step 4: Solve for $�$: $�-3=2�$ $�=\frac{�-3}{2}$

Step 5: Replace $�$ with ${�}^{-1}(�)$: ${�}^{-1}(�)=\frac{�-3}{2}$

So, the inverse function of $�(�)=2�+3$ is ${�}^{-1}(�)=\frac{�-3}{2}$.

You can use this inverse function to find the original input values when given the output values. For example, if you want to find ${�}^{-1}(5)$, you can substitute $�=5$ into the inverse function:

${�}^{-1}(5)=\frac{5-3}{2}=\frac{2}{2}=1$

So, ${�}^{-1}(5)=1$, meaning that if the original function $�(�)$ outputs 5, the inverse function ${�}^{-1}(�)$ inputs 1.

To find the inverse of a function and graph it, follow these steps:

Step 1: Start with the original function $�=�(�)$.

Step 2: Replace $�(�)$ with $�$:

• Replace $�(�)$ with $�$ to obtain the equation $�=\dots$.

Step 3: Swap the roles of $�$ and $�$:

• Replace $�$ with $�$ and $�$ with $�$ in the equation obtained from Step 2.

Step 4: Solve for $�$:

• Solve the equation from Step 3 for $�$ to express $�$ as a function of $�$. This will be your inverse function.

Step 5: Replace $�$ with ${�}^{-1}(�)$:

• Replace $�$ with ${�}^{-1}(�)$ to write the inverse function in the form ${�}^{-1}(�)=\dots$.

Step 6: Graph the inverse function:

• Plot the points on the graph of ${�}^{-1}(�)$ by swapping the coordinates of points on the graph of $�(�)$.
• Draw a smooth curve connecting these points to complete the graph of the inverse function.

Here's an example to illustrate these steps:

Example: Find the inverse of the function $�(�)=2�+3$, and graph the inverse function.

Step 1: Start with the original function, $�=2�+3$.

Step 2: Replace $�(�)$ with $�$: $�=2�+3$

Step 3: Swap the roles of $�$ and $�$: $�=2�+3$

Step 4: Solve for $�$: $�-3=2�$ $�=\frac{�-3}{2}$

Step 5: Replace $�$ with ${�}^{-1}(�)$: ${�}^{-1}(�)=\frac{�-3}{2}$

Step 6: Graph the inverse function: To graph ${�}^{-1}(�)$, you can plot points by swapping the coordinates of points on the graph of $�(�)$. For example, if $�(�)$ has a point (2, 7), then ${�}^{-1}(�)$ will have a corresponding point (7, 2). Do this for multiple points to create a smooth curve.

Now, you can create the graph of the inverse function ${�}^{-1}(�)$ based on the points you've identified and the equation ${�}^{-1}(�)=\frac{�-3}{2}$. The graph should be a straight line with a slope of 1/2 and a y-intercept of -3/2.

This is the process for finding the inverse of a function and graphing it. Remember that not all functions have inverses, and if an inverse exists, it must pass the horizontal line test, ensuring that no horizontal line intersects the graph more than once.

Chapter 3 Quiz

For the following exercises, determine whether each of the following relations is a function.

1. y=2x+8 2. {(2,1),(3,2),(−1,1),(0,−2)}

To determine whether a relation is a function, you can use the vertical line test. The vertical line test states that if a vertical line passes through a graph at more than one point, then the relation is not a function. If a vertical line passes through the graph at only one point (or doesn't intersect the graph at all), then the relation is a function.

Let's apply the vertical line test to the given relations:

1. y = 2x + 8: This is a linear equation representing a straight line. When you graph it, you'll see that for any given x-value, there's only one corresponding y-value. In other words, no vertical line will intersect the graph at more than one point. Therefore, this relation is a function.

2. {(2,1), (3,2), (-1,1), (0,-2)}: This is a set of ordered pairs. To determine if it's a function, consider the x-values. If there are no repeated x-values in the set (i.e., each x-value is unique), then it's a function. In this case, there are no repeated x-values: 2, 3, -1, and 0 are all different. Therefore, this relation is a function.

So, both of the given relations are functions.

For the following exercises, evaluate the function f(x)=−3x2+2x at the given input. 3. f(−2) 4. f(a) 5. Show that the function f(x)=−2(x−1)2+3 is not one-to-one. 6. Write the domain of the function f(x)=3−x−−−−√ in interval notation. 7. Given f(x)=2x2−5x, find f(a+1)−f(1) in simplest form. 8. Graph the function f(x)={x+1 if −x if −2<x<3x≥3 9. Find the average rate of change of the function f(x)=3−2x2+x by finding f(b)−f(a)b−a in simplest form.

Let's go through each of the exercises step by step:

3. 1. To evaluate $�(�)=-3{�}^2+2�$ at $�=-2$, simply substitute $-2$ for $�$:

   $�(-2)=-3(-2{)}^2+2(-2)$ $�(-2)=-3(4)-4$ $�(-2)=-12-4$ $�(-2)=-16$

   So, $�(-2)=-16$.

4. 2. To evaluate $�(�)=-3{�}^2+2�$ at $�=�$, simply substitute $�$ for $�$:

   $�(�)=-3{�}^2+2�$

5. 3. To show that the function $�(�)=-2(�-1{)}^2+3$ is not one-to-one, you can find two different values of $�$ that produce the same $�(�)$. For example, if $�=0$ and $�=2$, you get:

   $�(0)=-2(0-1{)}^2+3=-2(1{)}^2+3=-2+3=1$ $�(2)=-2(2-1{)}^2+3=-2(1{)}^2+3=-2+3=1$

   Both $�(0)$ and $�(2)$ equal 1, so the function is not one-to-one.

6. 4. The domain of the function $�(�)=3-\sqrt{�}$ consists of all the values of $�$ for which the expression under the square root is non-negative:

   $�\ge 0$

   So, in interval notation, the domain is $[0,\mathrm{\infty })$.

7. 5. Given $�(�)=2{�}^2-5�$, we can find $�(�+1)-�(1)$ as follows:

   $�(�+1)-�(1)=[2(�+1{)}^2-5(�+1)]-[2(1{)}^2-5(1)]$ $�(�+1)-�(1)=[2({�}^2+2�+1)-5�-5]-[2-5]$ $�(�+1)-�(1)=[2{�}^2+4�+2-5�-5]-[-3]$ $�(�+1)-�(1)=2{�}^2-�-3$

8. 6. Here's a description of the piecewise function $�(�)$:

   • For $�<-2$, $�(�)=�+1$.
   • For $-2\le �<3$, $�(�)=-�$.
   • For $�\ge 3$, $�(�)=3$.

9. 7. To find the average rate of change of the function $�(�)=3-2{�}^2+�$ from $�$ to $�$, use the formula:

   $\text{Average Rate of Change}=\frac{�(�)-�(�)}{�-�}$

10. 8. To find $(�\circ �)(�)$, first find $�(�)$ and then substitute it into $�(�)$. However, you haven't provided the functions $�(�)$ and $�(�)$, so I can't calculate this without that information.

11. 9. To find $(�\circ �)(1)$, follow the same steps as above, but substitute $�=1$ into the composite function. Again, you need the functions $�(�)$ and $�(�)$ to perform this calculation.

For the following exercises, use the functions f(x)=3−2x2+xand g(x)=x−−√ to find the composite functions. 10. (g∘f)(x) 11. (g∘f)(1)

To find the composite functions $(�\circ �)(�)$ and $(�\circ �)(1)$, you'll first need to find $�(�)$ and $�(�)$, and then perform the compositions.

Given:

• $�(�)=3-2{�}^2+�$
• $�(�)=\sqrt{�}$

Let's start with $(�\circ �)(�)$:

$(�\circ �)(�)=�(�(�))$

1. First, find $�(�)$: $�(�)=3-2{�}^2+�$

2. Now, substitute $�(�)$ into $�(�)$: $�(�(�))=\sqrt{3-2{�}^2+�}$

So, $(�\circ �)(�)=\sqrt{3-2{�}^2+�}$.

Now, let's find $(�\circ �)(1)$:

$(�\circ �)(1)=�(�(1))$

1. Find $�(1)$ by substituting $�=1$ into $�(�)$: $�(1)=3-2(1{)}^2+1=3-2+1=2$

2. Now, substitute $�(1)$ into $�(�)$: $�(�(1))=\sqrt{2}$

So, $(�\circ �)(1)=\sqrt{2}$.

The composite function $(�\circ �)(�)$ is $\sqrt{3-2{�}^2+�}$, and $(�\circ �)(1)=\sqrt{2}$.

12. Express H(x)=5x2−3x−−−−−−−√3 as a composition of two functions, f and g, where (f∘g)(x)=H(x).

To express $�(�)=5{�}^2-3\sqrt{3}$ as a composition of two functions $�$ and $�$ where $(�\circ �)(�)=�(�)$, we can break it down as follows:

Let $�(�)=5{�}^2$ and $�(�)=\sqrt{3-�}$.

Now, we want to find $�(�(�))$, which is the composition of $�$ and $�$. When we substitute $�(�)$ into $�(�)$, we get:

$�(�(�))=5(�(�){)}^2=5{\left(\sqrt{3-�}\right)}^2=5(3-�)=15-5�$

So, $(�\circ �)(�)=�(�)=15-5�$.

This means that we've expressed $�(�)$ as a composition of two functions: $�(�)=5{�}^2$ and $�(�)=\sqrt{3-�}$, where $(�\circ �)(�)=�(�)$.

For the following exercises, graph the functions by translating, stretching, and/or compressing a toolkit function. 13. f(x)=x+6−−−−√−1 14. f(x)=1x+2−1

To graph the given functions, it's helpful to identify the toolkit functions they are based on and apply transformations to them. The toolkit functions we'll use here are:

1. The square root function, $�(�)=\sqrt{�}$, which is commonly used as a toolkit function.
2. The reciprocal function, $�(�)=\frac{1}{�}$, which is also a common toolkit function.

Let's graph the given functions:

13. $�(�)=\sqrt{�+6}-1$

• Start with the square root toolkit function $�=\sqrt{�}$.
• Translate the graph to the left by 6 units to account for $�+6$.
• Shift the graph downward by 1 unit due to the "-1" at the end.

The graph of $�(�)=\sqrt{�+6}-1$ is the square root function translated 6 units to the left and 1 unit down from the origin.

14. $�(�)=\frac{1}{�+2}-1$

• Start with the reciprocal toolkit function $�=\frac{1}{�}$.
• Translate the graph to the left by 2 units to account for $�+2$.
• Shift the graph downward by 1 unit due to the "-1" at the end.

The graph of $�(�)=\frac{1}{�+2}-1$ is the reciprocal function translated 2 units to the left and 1 unit down from the origin.

Please note that the toolkit functions have specific characteristics, and transformations like translation and shifts modify those characteristics. For example, the square root function has a domain of $�\ge 0$ and the reciprocal function has a vertical asymptote at $�=0$. Transformations may affect these properties.

For the following exercises, determine whether the functions are even, odd, or neither. 15. f(x)=−5x2+9x6 16. f(x)=−5x3+9x5 17. f(x)=1x

To determine whether a function is even, odd, or neither, you can apply the following definitions:

1. Even Function: A function $�(�)$ is even if $�(-�)=�(�)$ for all $�$ in its domain.

2. Odd Function: A function $�(�)$ is odd if $�(-�)=-�(�)$ for all $�$ in its domain.

Let's apply these definitions to the given functions:

15. $�(�)=-5{�}^2+9{�}^6$

To check if it's even, substitute $-�$ for $�$:

$�(-�)=-5(-�{)}^2+9(-�{)}^6=-5{�}^2+9{�}^6$

Since $�(-�)=�(�)$, the function is even.

16. $�(�)=-5{�}^3+9{�}^5$

To check if it's odd, substitute $-�$ for $�$:

$�(-�)=-5(-�{)}^3+9(-�{)}^5=5{�}^3-9{�}^5$

Since $�(-�)=-�(�)$, the function is odd.

17. $�(�)=\frac{1}{�}$

This is a reciprocal function. To check if it's even or odd, let's substitute $-�$ for $�$:

$�(-�)=\frac{1}{-�}=-\frac{1}{�}$

Since $�(-�)=-�(�)$, the function is odd.

So, here are the conclusions:

• Function $�(�)=-5{�}^2+9{�}^6$ is an even function.
• Function $�(�)=-5{�}^3+9{�}^5$ is an odd function.
• Function $�(�)=\frac{1}{�}$ is an odd function.

18. Graph the absolute value function f(x)=−2|x−1|+3.

To graph the absolute value function $�(�)=-2\mathrm{\mid }�-1\mathrm{\mid }+3$, you can follow these steps:

1. Identify key points:

   • The vertex is at the point (1, 3).
   • The slope of the absolute value part is -2.

2. Plot the vertex:

   • Plot the point (1, 3), which is the vertex of the absolute value function.

3. Determine the direction of the "V":

   • Since the coefficient of the absolute value part is -2, the "V" shape of the graph will open downward.

4. Plot additional points:

   • Choose a few x-values on both sides of 1 to find additional points on the graph. You can use $�=0$, $�=2$, and $�=3$ for this purpose.

   When $�=0$: $�(0)=-2\mathrm{\mid }0-1\mathrm{\mid }+3=-2\mathrm{\mid }-1\mathrm{\mid }+3=-2(1)+3=1$

   So, one point is (0, 1).

   When $�=2$: $�(2)=-2\mathrm{\mid }2-1\mathrm{\mid }+3=-2\mathrm{\mid }1\mathrm{\mid }+3=-2(1)+3=1$

   So, another point is (2, 1).

   When $�=3$: $�(3)=-2\mathrm{\mid }3-1\mathrm{\mid }+3=-2\mathrm{\mid }2\mathrm{\mid }+3=-2(2)+3=-1$

   So, another point is (3, -1).

5. Draw the graph:

   • Connect the points you've plotted. The graph will be a "V" shape that opens downward with the vertex at (1, 3).

The graph of the absolute value function $�(�)=-2\mathrm{\mid }�-1\mathrm{\mid }+3$ will look like an inverted "V" centered at the point (1, 3), and it will be symmetric with respect to the vertical line $�=1$. The vertex is the highest point on the graph.

For the following exercises, find the inverse of the function. 19. f(x)=3x−5 20. f(x)=4x+7

To find the inverse of a function, $�(�)$, you can follow these steps:

1. Replace $�(�)$ with $�$: $�=�(�)$.
2. Swap the roles of $�$ and $�$: $�=�(�)$.
3. Solve for $�$ to express it as a function of $�$.

Let's find the inverses for the given functions:

19. $�(�)=3�-5$

20. Replace $�(�)$ with $�$: $�=3�-5$.

21. Swap the roles of $�$ and $�$: $�=3�-5$.

22. Solve for $�$: $3�=�+5$ $�=\frac{�+5}{3}$

So, the inverse of $�(�)$ is ${�}^{-1}(�)=\frac{�+5}{3}$.

20. $�(�)=4�+7$

21. Replace $�(�)$ with $�$: $�=4�+7$.

22. Swap the roles of $�$ and $�$: $�=4�+7$.

23. Solve for (y: $4�=�-7$ $�=\frac{�-7}{4}$

So, the inverse of $�(�)$ is ${�}^{-1}(�)=\frac{�-7}{4}$.