Problem 16 - College Algebra

Question

The functions in Exercises \(11-30\) are all one-to-one. For each function: a. Find an equation for \(f^{-1}(x),\) the inverse function. b. Verify that your equation is correct by showing that $$f\left(f^{-1}(x)\right)=x \text { and } f^{-1}(f(x))=x$$ $$f(x)=3 x-1$$

Step-by-Step Solution

Answer

The inverse of the function \(f(x) = 3x - 1\) is \(f^{-1}(x) = (x + 1) / 3\). The inverse function has been verified to be correct as both \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\) are proven true.

Step 1: Write the function y in terms of x

Firstly, the function is written in terms of \(y\), so \(f(x) = y = 3x - 1\).

Step 2: Solve given function for x

To find the inverse, solve this equation for \(x\). Add 1 to both sides to get \(y + 1 = 3x\), then divide both sides by 3 to isolate \(x\), resulting in \( x = (y + 1) / 3\).

Step 3: Express the inverse function

Replace \(y\) with \(x\) to get the inverse function \(f^{-1}(x) = (x + 1) / 3\).

Step 4: Verify the first equation \(f(f^{-1}(x)) = x\)

To verify the first equation, substitute \(f^{-1}(x)\) into \(f(x)\). So, \(f(f^{-1}(x)) = 3(f^{-1}(x)) - 1 = 3[(x + 1) / 3] - 1 = x + 1 - 1 = x.\) Thus, \(f(f^{-1}(x)) = x\) holds true.

Step 5: Verify the second equation \(f^{-1}(f(x)) = x\)

To verify the second equation, substitute \(f(x)\) into \(f^{-1}(x)\). So, \(f^{-1}(f(x)) = (f(x) + 1) / 3 = [(3x - 1) + 1] / 3 = (3x / 3) = x. \) Thus, \(f^{-1}(f(x)) = x\) also holds true.

Key Concepts

One-to-One Functions

One-to-one functions are special mathematical functions where each input (value from the domain) maps to a unique output (value in the range). This means that no two different inputs will produce the same output value. For a function to be one-to-one:

• Each element in the domain matches with a distinct element in the range.
• No two different domain elements map to the same range element.

A simple way to test if a function is one-to-one is to use the horizontal line test. If any horizontal line cuts the graph of the function more than once, then the function is not one-to-one. For functions like linear equations ( ax + b"), as long as the slope is not zero, these functions will be one-to-one.

Function Verification

To ensure we have correctly found the inverse of a function, we perform function verification. This step confirms the accuracy of our inversion process. To verify, we need to show that:

• The function and its inverse, when composed, yield the identity function.
• Specifically, if you compose the function with its inverse, either way, you get back the original input.

For the function \(f\) and its inverse \(f^{-1}\), you should verify:

• \(f(f^{-1}(x)) = x\)
• \(f^{-1}(f(x)) = x\)

If both compositions return the original input \(x\), then the inverse is verified. Thus, proper verification ensures both functions undo each other's operations.

Solving Equations

Solving an equation typically involves finding the variable values that make the equation true. When finding the inverse of a function, we ultimately solve for the original variable, using algebraic manipulations.

• First, identify and isolate the variable you want to solve in terms of another variable.
• Use inverse operations, such as adding/subtracting and multiplying/dividing, to move terms across the equation.

In the step-by-step solution, we see this process demonstrated as:

• Add 1 to both sides to balance the equation.
• Divide by 3 to isolate the variable \(x\).

These steps are common tools used to solve many types of equations involving linear functions.

Composition of Functions

Composition of functions involves combining two functions where the output of one function becomes the input for another. The notation \(f(g(x))\) reads as \(f\) composed with \(g\). Understanding composition is crucial for navigating the relationship between functions and their inverses.For inverses, the composition follows two important rules:

• \(f(f^{-1}(x)) = x\)
• \(f^{-1}(f(x)) = x\)

These rules ensure that when the functions are composed, they return the originating variable, confirming the correct inverse relationship. For example, in our given exercise, each step of substitution and simplification confirms these identities, illustrating the successful application of function compositions.