MTH120 College Algebra Chapter 5.7

 5.7 Inverses and Radical Functions:

Inverses:

In mathematics, the inverse of a function is a function that "undoes" the original function's operation. In other words, if you apply a function and then its inverse, you should get back to the original input.

For a function $�(�)$, its inverse is typically denoted as ${�}^{-1}(�)$. To find the inverse of a function, you can follow these general steps:

1. Replace $�(�)$ with $�$.
2. Swap the roles of $�$ and $�$, so you now have an equation with $�$ and $�$.
3. Solve for $�$.
4. Replace $�$ with ${�}^{-1}(�)$ to obtain the inverse function.

Not all functions have inverses. Inverses exist only for one-to-one functions, where each input corresponds to a unique output. Common examples of functions with inverses are linear functions and certain exponential and logarithmic functions.

Radical Functions:

A radical function is a type of function that involves a radical symbol (√), which represents a root. The most common radical function is the square root function, $�(�)=\sqrt{�}$, where $�$ must be non-negative since you cannot take the square root of a negative number in real numbers. Other examples include cube roots ($\sqrt[3]{�}$) and fourth roots ($\sqrt[4]{�}$).

Radical functions can also have additional transformations, such as:

1. Multiplication by a constant (e.g., $�\sqrt{�}$)
2. Addition or subtraction of constants (e.g., $\sqrt{�}+�$)
3. Combining with other functions (e.g., $�(�)=\sqrt{�(�)}$)

To work with radical functions, you should understand properties of radicals, such as simplifying expressions under the radical sign and solving equations involving radicals.

Finding the inverse of a polynomial function can be a bit more challenging than finding the inverse of simpler functions like linear or quadratic functions because not all polynomial functions have inverses. For an inverse to exist, the original polynomial function must be a one-to-one function, meaning that it passes the horizontal line test (each horizontal line intersects the graph at most once).

Here are the general steps to find the inverse of a polynomial function:

1. Start with the given polynomial function, which we'll denote as $�(�)$.
2. Replace $�(�)$ with $�$.
3. Swap the roles of $�$ and $�$, so you now have an equation with $�$ and $�$.

The equation you have should look like this:

$�=�(�)$

4. Now, solve for $�$ in terms of $�$. This means you want to isolate $�$ on one side of the equation.

5. Once you have $�$ expressed as a function of $�$, you can denote this as $�=�(�)$. This is your inverse function.

6. Finally, swap $�$ and $�$ in your equation to express the inverse function as $�=�(�$.

Keep in mind that the complexity of finding the inverse depends on the degree and nature of the polynomial. For simple linear and quadratic polynomials, the process is straightforward. For higher-degree polynomials, it can be more complex and might involve solving for $�$ using algebraic techniques or calculus.

Let's illustrate this process with an example:

Example: Finding the Inverse of a Quadratic Polynomial

Given the quadratic polynomial $�(�)=�{�}^2+��+�$, where $�\mathrm{\ne }0$, we want to find its inverse.

1. Start with $�(�)$: $�=�{�}^2+��+�$
2. Swap $�$ and $�$ to get: $�=�{�}^2+��+�$
3. Solve for $�$: $�{�}^2+��+�-�=0$
4. You can use the quadratic formula to solve for $�$: $�=\frac{-�\pm \sqrt{{�}^2-4��+4��}}{2�}$

Now, $�=�(�)$, where $�(�)$ is the inverse function of $�(�)$. Note that in this case, you'll have two possible values for $�$ due to the ± sign. This is because the quadratic function is not always one-to-one, and the inverse may not be unique for all inputs.

Keep in mind that not all polynomial functions have inverses, especially if they are not one-to-one functions. Additionally, the resulting inverse might not always be a polynomial, depending on the original polynomial function's nature.

To verify that two functions are inverses of each other, you need to check if composing them in both orders results in the identity function. In other words, you need to verify that:

1. If $�(�)$ is a function, and $�(�)$ is its supposed inverse, then $�(�(�))=�$ for all valid values of $�$.
2. Similarly, $�(�(�))=�$ for all valid values of $�$.

Here are two examples of verifying functions as inverses of each other:

Example 1: Linear Functions

Let's take two linear functions:

$�(�)=2�+3$ and $�(�)=\frac{�-3}{2}$.

To check if $�(�)$ and $�(�)$ are inverses of each other:

1. Calculate $�(�(�))$:

   $�(�(�))=2\left(\frac{�-3}{2}\right)+3$

   Simplify:

   $�(�(�))=(�-3)+3=�$

   This shows that $�(�(�))$ is equal to the identity function, so $�(�)$ and $�(�)$ are inverses.

2. Calculate (g(f(x)):

   $�(�(�))=\frac{2�+3-3}{2}$

   Simplify:

   $�(�(�))=\frac{2�}{2}=�$

   This also shows that $�(�(�))$ is equal to the identity function, confirming that $�(�)$ and $�(�)$ are inverses of each other.

Example 2: Exponential and Logarithmic Functions

Consider the following functions:

$�(�)={�}^{�}$ and $�(�)=\ln (�)$.

To check if $�(�)$ and $�(�)$ are inverses of each other:

1. Calculate $�(�(�))$:

   $�(�(�))={�}^{\ln (�)}$

   Using the fact that $\ln (�)$ and ${�}^{�}$ are inverse operations:

   $�(�(�))=�$

   So, $�(�)$ and $�(�)$ are inverses.

2. Calculate (g(f(x)):

   $�(�(�))=\ln ({�}^{�})$

   Again, using the fact that $\ln (�)$ and ${�}^{�}$ are inverse operations:

   $�(�(�))=�$

   This also confirms that $�(�)$ and $�(�)$ are inverses of each other.

In both examples, we've shown that composing the functions in both orders results in the identity function, which verifies that they are indeed inverses of each other.

To find the inverse of a polynomial function by restricting the domain to make it one-to-one, you can follow these steps:

1. Start with the Polynomial Function: Begin with the given polynomial function, which we'll denote as $�(�)$. You want to find its inverse.

2. Determine the Domain: Analyze the original function $�(�)$ and identify the portion of its domain where it is one-to-one. This usually involves selecting a continuous interval where the function is strictly increasing or decreasing, thus passing the horizontal line test.

3. Restrict the Domain: Define a new function with the same equation as $�(�)$, but with the domain restricted to the identified interval. This restricted function is now one-to-one on that interval.

4. Swap Roles of $�$ and $�$: Replace $�(�)$ with $�$ and rewrite the restricted function with $�$ and $�$ swapped.

5. Solve for $�$: Solve the new equation for $�$ in terms of $�$. This will give you the expression for the inverse function.

6. Swap Roles of $�$ and $�$ Again: Finally, replace $�$ with ${�}^{-1}(�)$ to express the inverse function.

Let's illustrate this process with an example:

Example: Finding the Inverse of a Polynomial Function by Restricting the Domain

Given the polynomial function $�(�)={�}^3+2�$ and you want to find its inverse by restricting the domain to make it one-to-one.

1. Start with $�(�)$: $�={�}^3+2�$.

2. Determine the domain: The function $�(�)$ is continuous and strictly increasing for all real numbers, so it is already one-to-one over its entire domain. In this case, there's no need to restrict the domain.

3. Swap roles of $�$ and $�$: $�={�}^3+2�$.

4. Solve for $�$: ${�}^3+2�-�=0$

5. Swap roles of $�$ and $�$ again: The inverse function is ${�}^{-1}(�)$.

The resulting inverse function, in this case, is not easily expressed in terms of elementary functions. It's a cubic equation in $�$. It might be challenging to find an explicit formula for the inverse function, but you can represent it numerically or graphically.

Keep in mind that not all polynomial functions can have easily expressible inverses, especially for higher-degree polynomials. The existence and nature of an inverse depend on the polynomial's properties, including whether it's one-to-one over its entire domain.

To find the inverse of a polynomial function by restricting the domain to make it one-to-one, you need to choose a specific interval within the domain where the polynomial is one-to-one. This interval should be such that the polynomial is strictly increasing or strictly decreasing over that range. Here are the steps to do this:

Step 1: Start with the Polynomial Function

Begin with the given polynomial function, which we'll denote as $�(�)$. You want to find its inverse.

Step 2: Determine the Interval

Analyze the original function $�(�)$ to identify an interval in its domain where it is one-to-one. This usually involves selecting a continuous interval where the function is strictly increasing or strictly decreasing. For a polynomial to be one-to-one, it must pass the horizontal line test.

Step 3: Restrict the Domain

Define a new function with the same equation as $�(�$), but with the domain restricted to the identified interval. This restricted function is now one-to-one on that interval.

Step 4: Swap Roles of $�$ and (y)

Replace $�(�)$ with $�$ and rewrite the restricted function with $�$ and $�$ swapped.

Step 5: Solve for $�$

Solve the new equation for $�$ in terms of $�$. This will give you the expression for the inverse function.

Step 6: Swap Roles of $�$ and (y) Again

Finally, replace $�$ with ${�}^{-1}(�)$ to express the inverse function.

Let's illustrate this process with an example:

Example: Finding the Inverse of a Quadratic Polynomial by Restricting the Domain

Given the quadratic polynomial function $�(�)={�}^2-4$, we want to find its inverse by restricting the domain to make it one-to-one.

Step 1: Start with $�(�)$

$�={�}^2-4$.

Step 2: Determine the Interval

This quadratic function is a parabola that opens upward. To restrict the domain and make it one-to-one, we can choose the interval $[-2,\mathrm{\infty })$, where the function is strictly increasing. In other words, $�\ge -2$.

Step 3: Restrict the Domain

Define a new function ${�}_1(�)$ with the same equation as $�(�)$ but restricted to the interval $[-2,\mathrm{\infty })$:

${�}_1(�)={�}^2-4$, where $�\ge -2$.

Step 4: Swap Roles of $�$ and (y)

$�={�}^2-4$.

Step 5: Solve for $�$

${�}^2=�+4$

$�=\pm \sqrt{�+4}$

Step 6: Swap Roles of $�$ and (y) Again

The inverse function, ${�}^{-1}(�)$, is:

${�}^{-1}(�)=\pm \sqrt{�+4}$

It's important to note that since we have $\pm$ in the expression, the inverse is a multivalued function. Depending on the choice of sign (positive or negative), you get two different branches of the inverse. These branches may not satisfy the horizontal line test over the entire domain, but they do when restricted to the specified interval $[-2,\mathrm{\infty })$.

Restricting the domain of a function is a common technique in mathematics, particularly when you want to make a function one-to-one or simplify its behavior within a specific range. Here's how you can go about restricting the domain:

1. Start with the Original Function: Begin with the given function, which we'll denote as $�(�)$. This is the function you want to restrict.

2. Determine the New Domain: Identify the interval or range within which you want to restrict the function. This interval should be carefully chosen based on your goals. You might want to make the function one-to-one, simplify its behavior, or focus on a specific part of the function's graph.

3. Express the Restriction: Define a new function, ${�}_1(�)$, with the same equation as $�(�)$ but with a restricted domain. This means you'll indicate the domain explicitly in the function definition.

For example, if you want to restrict the function to the interval $[�,�]$, the restricted function would be:

${�}_1(�)=�(�)\text{ for }�\le �\le �$

4. Verify One-to-One (If Intended): If the purpose of restricting the domain is to make the function one-to-one, you need to check whether the restricted function indeed passes the horizontal line test (it should not intersect a horizontal line more than once). If it doesn't pass the test, you might need to choose a different interval.

5. Use the Restricted Function: The restricted function, ${�}_1(�)$, now behaves only within the specified domain. You can work with this function as needed.

Here's a simple example:

Example: Restricting the Domain of a Piecewise Function

Suppose you have a piecewise function $�(�)$ defined as follows:

x^2, & \text{if } x \geq 0 \\ 2x - 1, & \text{if } x < 0 \end{cases}\] Now, let's say you want to restrict the domain to the interval \([0, \infty)\) to focus on the behavior of the function when \(x\) is non-negative. **1. Start with the Original Function:** \(f(x)\) as defined above. **2. Determine the New Domain:** The interval \([0, \infty)\) is chosen. **3. Express the Restriction:** Define a new function, \(f_1(x)\), with the restricted domain: \[f_1(x) = \begin{cases} x^2, & \text{if } x \geq 0 \end{cases}\] **4. Verify One-to-One (If Intended):** In this case, we don't need to worry about making it one-to-one, as the restricted portion of the function (the quadratic part) is already one-to-one for \(x \geq 0\). **5. Use the Restricted Function:** You can now work with the restricted function \(f_1(x)\), which is \(x^2\) for \(x \geq 0\). This makes it easier to analyze and understand the behavior of the function in the non-negative domain.

Restricting the domain of a polynomial function that is not one-to-one and finding the inverse can be challenging because, by definition, the inverse of a function exists only when it is one-to-one. However, if you restrict the domain to a portion of the polynomial function where it is one-to-one, you can find the inverse of that restricted portion.

Let's go through this process step by step with an example:

Example: Restricting the Domain of a Polynomial Function

Consider the polynomial function $�(�)={�}^2$ for all real numbers. This function is not one-to-one because multiple values of $�$ can result in the same $�$ value. To make it one-to-one, you can restrict the domain to the interval $[0,\mathrm{\infty })$, where the function is strictly increasing.

Step 1: Start with $�(�)$:

$�(�)={�}^2$.

Step 2: Determine the New Domain:

Choose the interval $[0,\mathrm{\infty })$ to restrict the function.

Step 3: Express the Restriction:

Define a new function ${�}_1(�)$ with the same equation as $�(�)$ but with the restricted domain:

${�}_1(�)={�}^2\text{ for }�\ge 0$

Step 4: Verify One-to-One (If Intended):

Since the restricted portion is ${�}^2$ for $�\ge 0$, it's already one-to-one in this interval.

Step 5: Use the Restricted Function:

You can now work with the restricted function ${�}_1(�)={�}^2$ for $�\ge 0$. This function is one-to-one within the specified domain.

Step 6: Find the Inverse of the Restricted Function:

To find the inverse of the restricted function ${�}_1(�)={�}^2$ for $�\ge 0$, follow the standard procedure:

1. Start with ${�}_1(�)$: $�={�}^2$.
2. Swap $�$ and $�$: $�={�}^2$.
3. Solve for $�$: $�=\sqrt{�}$.
4. Swap $�$ and $�$ again: The inverse function is ${�}_1^{-1}(�)=\sqrt{�}$.

Keep in mind that this process gives you the inverse of the restricted portion of the polynomial function within the specified domain. The inverse function in this case is ${�}_1^{-1}(�)=\sqrt{�}$ for $�\ge 0$.

Solving applications of radical functions involves using radical expressions to solve real-world problems. Radical functions typically involve square roots, cube roots, or other types of roots. To solve these types of applications, follow these general steps:

1. Read the Problem Carefully: Understand the problem statement, including what's being asked and any given information or constraints.

2. Define Variables: Identify the quantities that need to be found or related, and define variables to represent them. Often, you will use $�$ for the unknown value.

3. Set Up an Equation: Translate the information given in the problem into a mathematical equation involving radical expressions. This equation should represent the relationship between the variables.

4. Solve the Equation: Use algebraic techniques to solve the equation. For radical equations, you typically need to isolate the radical expression on one side and then square (or cube, etc.) both sides to eliminate the radical.

5. Check for Extraneous Solutions: Be aware that some solutions may be extraneous, meaning they don't satisfy the original problem. For example, solutions that lead to negative values inside a square root may be extraneous if not relevant to the real-world context.

6. Interpret the Solution: Provide the solution in the context of the problem. Make sure to answer the original question asked in the problem statement.

Let's look at an example of solving an application of a radical function:

Example: Finding the Distance

Suppose you want to find the distance between two points $({�}_1,{�}_1)$ and $({�}_2,{�}_2)$ in a two-dimensional coordinate system. The distance is given by the formula:

$�=\sqrt{({�}_2-{�}_1{)}^2+({�}_2-{�}_1{)}^2}$

Suppose you have two points: $(2,3)$ and $(5,7)$, and you want to find the distance between them.

Step 1: Read the Problem You want to find the distance between two points.

Step 2: Define Variables Let $�$ represent the distance, ${�}_1=2$, ${�}_1=3$, ${�}_2=5$, and ${�}_2=7$.

Step 3: Set Up an Equation Use the distance formula: $�=\sqrt{(5-2{)}^2+(7-3{)}^2}$

Step 4: Solve the Equation Calculate the values inside the square root: $�=\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5$

Step 5: Check for Extraneous Solutions In this case, there are no extraneous solutions.

Step 6: Interpret the Solution The distance between the points $(2,3)$ and $(5,7)$ is 5 units.

This is a simple example of solving an application of a radical function. In more complex problems, you might encounter cubic roots or other types of radicals, but the general approach remains the same: translate the problem into a mathematical equation, solve it, and interpret the results in the context of the problem.

To find the inverse of a radical function, you can follow a similar process to finding the inverse of any function. Here are the general steps:

1. Start with the Radical Function: Begin with the given radical function, which we'll denote as $�(�)$. This is the function for which you want to find the inverse.

2. Replace $�(�)$ with $�$: Let $�=�(�)$. This is done to set up the equation for finding the inverse.

3. Swap $�$ and $�$: Swap the roles of $�$ and $�$ in the equation, which gives you an equation in terms of $�$ and $�$.

4. Solve for $�$: Solve the equation for $�$ in terms of $�$. This will give you an expression for the inverse function.

**5. Replace $�$ with f^{-1}(x):** Replace \(y with ${�}^{-1}(�)$ to express the inverse function.

Let's illustrate this process with an example:

Example: Finding the Inverse of a Square Root Function

Given the function $�(�)=\sqrt{�}$, you want to find its inverse.

Step 1: Start with $�(�)$: $�=\sqrt{�}$

Step 2: Replace $�(�)$ with $�$: $�=\sqrt{�}$

Step 3: Swap $�$ and $�$: $�=\sqrt{�}$

Step 4: Solve for $�$: Square both sides to isolate $�$: ${�}^2=�$

**Step 5: Replace $�$ with f^{-1}(x):** The inverse function is \(f^{-1}(x) = x^2.

So, the inverse of the square root function is the square function ${�}^{-1}(�)={�}^2$. You can check this by composing the original function and its inverse to see that they undo each other, i.e., $�({�}^{-1}(�))=�$ and ${�}^{-1}(�(�))=�$, where $�(�)=\sqrt{�}$ and ${�}^{-1}(�)={�}^2$.

Solving applications of radical functions involves using radical expressions to solve real-world problems. Radical functions typically involve square roots, cube roots, or other types of roots. To solve these types of applications, follow these general steps:

1. Read the Problem Carefully: Understand the problem statement, including what's being asked and any given information or constraints.

2. Define Variables: Identify the quantities that need to be found or related, and define variables to represent them. Often, you will use $�$ for the unknown value.

3. Set Up an Equation: Translate the information given in the problem into a mathematical equation involving radical expressions. This equation should represent the relationship between the variables.

4. Solve the Equation: Use algebraic techniques to solve the equation. For radical equations, you typically need to isolate the radical expression on one side and then square (or cube, etc.) both sides to eliminate the radical.

5. Check for Extraneous Solutions: Be aware that some solutions may be extraneous, meaning they don't satisfy the original problem. For example, solutions that lead to negative values inside a square root may be extraneous if not relevant to the real-world context.

6. Interpret the Solution: Provide the solution in the context of the problem. Make sure to answer the original question asked in the problem statement.

Let's look at an example of solving an application of a radical function:

Example: Finding the Distance

Suppose you want to find the distance between two points $({�}_1,{�}_1)$ and $({�}_2,{�}_2)$ in a two-dimensional coordinate system. The distance is given by the formula:

$�=\sqrt{({�}_2-{�}_1{)}^2+({�}_2-{�}_1{)}^2}$

Suppose you have two points: $(2,3)$ and ((5, 7), and you want to find the distance between them.

Step 1: Read the Problem You want to find the distance between two points.

Step 2: Define Variables Let $�$ represent the distance, ${�}_1=2$, ${�}_1=3$, ${�}_2=5$, and ${�}_2=7$.

Step 3: Set Up an Equation Use the distance formula: $�=\sqrt{(5-2{)}^2+(7-3{)}^2}$

Step 4: Solve the Equation Calculate the values inside the square root: $�=\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5$

Step 5: Check for Extraneous Solutions In this case, there are no extraneous solutions.

Step 6: Interpret the Solution The distance between the points $(2,3)$ and $(5,7)$ is 5 units.

This is a simple example of solving an application of a radical function. In more complex problems, you might encounter cubic roots or other types of radicals, but the general approach remains the same: translate the problem into a mathematical equation, solve it, and interpret the results in the context of the problem.

To determine the domain of a radical function composed with other functions, you'll need to consider the domains of all the individual functions involved in the composition and find the intersection of those domains. The domain of the composite function is the set of values that makes each function in the composition valid. Here are the steps:

1. Identify the Composite Function: Start by identifying the composite function, which involves a radical function and one or more other functions. For example, you might have a composite function like $�(�(�))$, where $�(�)$ is a radical function.

2. Determine the Domain of the Inner Function: Identify the innermost function, which in this case is $�(�)$. Determine the domain of $�(�)$ by considering any restrictions imposed by the radical expression. For example, if $�(�)=\sqrt{�}$, the domain of $�(�)$ would be $�\ge 0$ since you cannot take the square root of a negative number.

3. Determine the Domain of Other Functions: If there are other functions involved in the composition (e.g., $�(�)$ in $�(�(�))$), determine the domain of these functions separately. Each function in the composition should have its domain considered.

4. Intersect Domains: Find the intersection of the domains of all functions involved. The domain of the composite function will be the set of values that satisfy all the domain restrictions of the functions involved. This means you are looking for the values of $�$ that make each function in the composition valid.

5. Express the Domain: Once you've found the intersection of the domains, express the domain of the composite function in terms of inequalities or intervals. This will specify the valid values of $�$ for the entire composition.

Let's look at an example:

Example: Determining the Domain of $�(�(�))$

Suppose you have the composite function $�(�(�))$ where $�(�)=\sqrt{�}$ and $�(�)=\frac{1}{�}$.

1. Identify the Composite Function: You have $�(�(�))$.

2. Determine the Domain of $�(�)$: For $�(�)=\sqrt{�}$, the domain is $�\ge 0$ because the square root is defined only for non-negative values.

3. Determine the Domain of $�(�)$: For $�(�)=\frac{1}{�}$, the domain is $�\mathrm{\ne }0$, as you cannot divide by zero.

4. Intersect Domains: The domain of $�(�(�))$ is the intersection of the domains of $�(�)$ and $�(�)$. In this case, the valid values of $�$ are $�\ge 0$ and $�\mathrm{\ne }0$.

5. Express the Domain: The domain of $�(�(�))$ is $�>0$, which means it includes all positive real numbers but excludes zero.

In this example, the domain of the composite function $�(�(�))$ is all positive real numbers. This is because you have to consider the domains of both $�(�)$ and $�(�)$ and take the intersection of their domains.

To find the inverse of a rational function, you typically follow these steps. Keep in mind that not all rational functions have inverses. The existence of an inverse depends on whether the original function is one-to-one.

Step 1: Start with the Rational Function

Begin with the given rational function, which we'll denote as $�(�)$. You want to find its inverse.

*Step 2: Replace $�(�)$ with (y*

Let $�=�(�)$. This step sets up the equation for finding the inverse.

*Step 3: Swap $�$ and (y*

Swap the roles of $�$ and $�$ in the equation, creating an equation in terms of $�$ and $�$.

*Step 4: Solve for (y*

Solve the equation for $�$ in terms of $�$. This will give you an expression for the inverse function.

*Step 5: Replace $�$ with (f^{-1}(x)*

Finally, replace $�$ with ${�}^{-1}(�)$ to express the inverse function.

Let's illustrate this process with an example:

Example: Finding the Inverse of a Rational Function

Given the rational function $�(�)=\frac{�-1}{2�+3}$, you want to find its inverse.

*Step 1: Start with (f(x)*

$�=\frac{�-1}{2�+3}$

*Step 2: Replace (f(x)\ with (y*

$�=\frac{�-1}{2�+3}$

*Step 3: Swap $�$ and (y*

$�=\frac{�-1}{2�+3}$

*Step 4: Solve for (y*

Let's solve for $�$:

Multiply both sides by $(2�+3)$ to eliminate the fraction:

$�(2�+3)=�-1$

Distribute $�$ on the left side:

$2��+3�=�-1$

Now, move all terms involving $�$ to one side:

$2��-�=-3�+1$

Factor out $�$ on the left side:

$�(2�-1)=-3�+1$

Finally, solve for $�$:

$�=\frac{-3�+1}{2�-1}$

*Step 5: Replace $�$ with (f^{-1}(x)*

The inverse function is ${�}^{-1}(�)=\frac{-3�+1}{2�-1}$.

This is the inverse of the given rational function $�(�)$. Make sure to check that the original function and its inverse satisfy the properties of inverses, which include composing them to get the identity function and ensuring the domain and range of the inverse match the range and domain of the original function, respectively.

Finding the inverse of a rational function involves swapping the roles of $�$ and $�$, solving for $�$, and then replacing $�$ with ${�}^{-1}(�)$. Let's work through an example to illustrate this process:

Example: Finding the Inverse of a Rational Function

Given the rational function $�(�)=\frac{�+2}{3�-1}$, find its inverse.

Step 1: Start with $�(�)$:

$�=\frac{�+2}{3�-1}$

Step 2: Replace $�(�)$ with $�$:

$�=\frac{�+2}{3�-1}$

Step 3: Swap $�$ and $�$:

$�=\frac{�+2}{3�-1}$

Step 4: Solve for $�$:

Now, let's solve for $�$. To do this, we'll isolate (y:

Multiply both sides by $3�-1$ to eliminate the fraction:

$�(3�-1)=�+2$

Distribute $�$ on the left side:

$3��-�=�+2$

Move terms involving $�$ to the left side:

$3��-�=�+2$

Factor out $�$ on the left side:

$�(3�-1)=�+2$

Finally, solve for $�$:

$�=\frac{�+2}{3�-1}$

Step 5: Replace $�$ with (f^{-1}(x):

The inverse function is ${�}^{-1}(�)=\frac{�+2}{3�-1}$.

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

Make sure to check that the original function and its inverse satisfy the properties of inverses, such as composing them to get the identity function and ensuring the domain and range of the inverse match the range and domain of the original function, respectively.

In this example, we found the inverse of a rational function, which is simply the same function with $�$ and $�$ swapped in its expression.

For the following exercises, find the inverse of the function on the given domain. 1. f(x)=(x−6)2, [4,∞) 2. f(x)=(x+3)2, [−2,∞) 3. f(x)=(x+7)2−3, [−1,∞) 4. f(x)=2x2+5,(∞,0] 5. f(x)=11−x2, [0,∞) 6. f(x)=9−x2, [0,∞)

To find the inverse of each function on the given domain, follow these general steps:

1. Start with the given function, $�(�)$.
2. Replace $�(�)$ with $�$.
3. Swap $�$ and $�$ to get an equation in terms of $�$ and $�$.
4. Solve for $�$ in terms of $�$.
5. Replace $�$ with ${�}^{-1}(�)$ to express the inverse function.

Let's apply these steps to the provided functions:

1. $�(�)=(�-6{)}^2,[4,\mathrm{\infty })$:

• Start with $�(�)$: $�=(�-6{)}^2$.
• Swap $�$ and $�$: $�=(�-6{)}^2$.
• Solve for $�$: $�=\sqrt{�}+6$.
• Replace $�$ with ${�}^{-1}(�)$: ${�}^{-1}(�)=\sqrt{�}+6$.

2. $�(�)=(�+3{)}^2,[-2,\mathrm{\infty })$:

• Start with $�(�)$: $�=(�+3{)}^2$.
• Swap $�$ and $�$: $�=(�+3{)}^2$.
• Solve for (y: $�=\sqrt{�}-3$.
• Replace $�$ with ${�}^{-1}(�)$: ${�}^{-1}(�)=\sqrt{�}-3$.

3. $�(�)=(�+7{)}^2-3,[-1,\mathrm{\infty })$:

• Start with $�(�)$: $�=(�+7{)}^2-3$.
• Swap $�$ and $�$: $�=(�+7{)}^2-3$.
• Solve for (y: $�=\sqrt{�+3}-7$.
• Replace $�$ with ${�}^{-1}(�)$: ${�}^{-1}(�)=\sqrt{�+3}-7$.

4. $�(�)=2{�}^2+5,(-\mathrm{\infty },0]$:

• Start with $�(�)$: $�=2{�}^2+5$.
• Swap $�$ and $�$: $�=2{�}^2+5$.
• Solve for (y: $�=\sqrt{\frac{�-5}{2}}$.
• Replace $�$ with ${�}^{-1}(�)$: ${�}^{-1}(�)=\sqrt{\frac{�-5}{2}}$.

5. $�(�)=11-{�}^2,[0,\mathrm{\infty })$:

• Start with $�(�)$: $�=11-{�}^2$.
• Swap $�$ and $�$: $�=11-{�}^2$.
• Solve for (y: $�=\sqrt{11-�}$.
• Replace $�$ with ${�}^{-1}(�)$: ${�}^{-1}(�)=\sqrt{11-�}$.

6. $�(�)=9-{�}^2,[0,\mathrm{\infty })$:

• Start with $�(�)$: $�=9-{�}^2$.
• Swap $�$ and $�$: $�=9-{�}^2$.
• Solve for (y: $�=\sqrt{9-�}$.
• Replace $�$ with ${�}^{-1}(�)$: ${�}^{-1}(�)=\sqrt{9-�}$.

These are the inverse functions for the given original functions on the specified domains. Make sure to double-check your work and verify that the inverses behave as expected.