Problem 21 - Intermediate Algebra

Question

Find the inverse of the given function by using the "undoing process," and then verify that \(\left(f \circ f^{-1}\right)(x)=x\) and \(\left(f^{-1} \circ f\right)(x)=x\). (Objective 4) $$f(x)=5 x-4$$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \frac{x + 4}{5} \). Verification shows \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

Step 1: Write the function equation

Start with the given function equation: \[ y = 5x - 4 \] This represents the function \( f(x) = 5x - 4 \).

Step 2: Swap x and y

To find the inverse function, interchange \(x\) and \(y\): \[ x = 5y - 4 \].

Step 3: Solve for y

Focus on isolating \(y\) by solving the equation: \[ x + 4 = 5y \] Then, divide by 5 to obtain: \[ y = \frac{x + 4}{5} \].

Step 4: Write the inverse function

The inverse function of \( f(x) = 5x - 4 \) is: \[ f^{-1}(x) = \frac{x + 4}{5} \].

Step 5: Verify \( \left(f \circ f^{-1}\right)(x) = x \)

Substitute \( f^{-1}(x) \) into \( f(x) \): \[ f(f^{-1}(x)) = 5\left(\frac{x + 4}{5}\right) - 4 \] Simplify:\[ = x + 4 - 4 \] \[ = x \].

Step 6: Verify \( \left(f^{-1} \circ f\right)(x) = x \)

Substitute \( f(x) \) into \( f^{-1}(x) \): \[ f^{-1}(f(x)) = \frac{5x - 4 + 4}{5} \] Simplify:\[ = \frac{5x}{5} \] \[ = x \].

Key Concepts

Undoing Process

The concept of the "undoing process" is essentially about reversing the steps of a function to find its inverse. Think of a function as a series of actions that transform an input to an output. The inverse function undoes these actions to return to the original input. For the function given as \( f(x) = 5x - 4 \), finding the inverse involves reversing these operations.
Here is a simple breakdown:

• The original operation is to multiply by 5 and then subtract 4.
• To "undo" this, you start by adding 4 (opposite of subtracting 4), which brings us one step back.
• Next, you divide by 5, which reverses the multiplication by 5.
• Thus, the "undoing process" effectively is: add 4, then divide by 5.

This sequence helps us recover the original input value, effectively finding the inverse.

Function Composition

Function composition is a way of combining two functions wherein the output of one function becomes the input of another. When verifying inverse functions, function composition allows us to check if one function truly undoes the other.
In mathematical terms, if \( g(x) \) is the inverse of \( f(x) \), then composing \( f \) with \( g \) and vice versa will return \( x \), our original input:

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

For our function \( f(x) = 5x - 4 \) and its inverse \( f^{-1}(x) = \frac{x + 4}{5} \), we can check this by substituting one into the other, confirming that both compositions equal \( x \). This step validates that the two functions are indeed inverses.

Interchange Variables

Finding an inverse function often involves swapping variables, a concept known as interchanging variables. This step is crucial because to reverse the output-to-input relationship, we assign the roles of inputs and outputs to each other.
To illustrate:

• Start with the function \( y = 5x - 4 \), where \( y \) is the output and \( x \) is the input.
• Interchange them: Let \( x \) become the output and \( y \) the new input, resulting in: \( x = 5y - 4 \).
• After interchanging, solve for \( y \) to express the inverse: \( y = \frac{x + 4}{5} \).

This swapping trick simplifies the process of determining an inverse. It provides a framework for reversing the direction between inputs and outputs in the context of functions.