Problem 27 - 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) $$f(x)=\frac{1}{2} x+4$$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = 2(x - 4) \). Both compositions verify identity: \( (f \circ f^{-1})(x) = x \) and \( (f^{-1} \circ f)(x) = x \).

Step 1: Understand the Function

The function given is \( f(x) = \frac{1}{2}x + 4 \). To find its inverse, we need to express \( x \) in terms of \( y \), where \( y = f(x) \).

Step 2: Express y in terms of x

We start by setting \( y = f(x) = \frac{1}{2}x + 4 \). Our goal is to solve for \( x \) in terms of \( y \).

Step 3: Swap and solve for x

Rewrite the equation as \( y = \frac{1}{2}x + 4 \). To isolate \( x \), first subtract 4 from both sides to get \( y - 4 = \frac{1}{2}x \).

Step 4: Isolate x

Multiply both sides of the equation \( y - 4 = \frac{1}{2}x \) by 2 to solve for \( x \), yielding \( x = 2(y - 4) \).

Step 5: Express inverse function

Replace \( y \) with \( x \) to write the inverse function. Thus, \( f^{-1}(x) = 2(x - 4) \).

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

Substitute \( f^{-1}(x) = 2(x - 4) \) into the function \( f \), giving \( f(f^{-1}(x)) = f(2(x - 4)) = \frac{1}{2}(2(x - 4)) + 4 = (x - 4) + 4 = x \). This confirms that \( (f \circ f^{-1})(x) = x \).

Step 7: Verify \( (f^{-1} \circ f)(x) = x \)

Substitute \( f(x) = \frac{1}{2}x + 4 \) into the inverse function \( f^{-1}(x) \), giving \( f^{-1}(f(x)) = f^{-1}(\frac{1}{2}x + 4) = 2((\frac{1}{2}x + 4) - 4) = 2(\frac{1}{2}x) = x \). This confirms that \( (f^{-1} \circ f)(x) = x \).

Key Concepts

Function Composition

Function composition is all about combining two functions together. Imagine you have two functions, \( f \) and \( g \). When you compose them, you create a new function, like \( (f \circ g) \), which means you first apply \( g \) to an input and then apply \( f \) to the result of \( g \). This method effectively chains two functions together to act as one.

• Think of \( (f \circ g)(x) \) as a two-step process: \( g \) does something to the input \( x \), then \( f \) works on the outcome of \( g(x) \).
• In the context of inverses, function composition can help verify if two functions are indeed inverses of each other.

In our example, we checked \( (f \circ f^{-1})(x) = x \) and \( (f^{-1} \circ f)(x) = x \) to confirm that they undo each other's actions. When successful, both compositions work to return the original input value \( x \). This indicates a perfect inverse relationship between \( f \) and \( f^{-1} \).
By understanding how function composition operates, you can better grasp the interactions between functions and how they can be manipulated.

Undoing Process

The undoing process is a great way to find the inverse of a function. Essentially, it involves reversing the operations in a given function to express the input variable, typically \( x \), in terms of the output variable, usually \( y \). This reversal effectively "undoes" the function, allowing you to find the inverse.

• Start by expressing \( y = f(x) \) based on your original function. In our example, that was \( y = \frac{1}{2}x + 4 \).
• Next, rearrange the equation to solve \( x \) in terms of \( y \). This may involve steps like adding, subtracting, multiplying, or dividing.
• Finally, swap \( y \) with \( x \) to write down the inverse function. The inverse function represents how the original function's output can be used to regain the initial input.

For \( f(x) = \frac{1}{2}x + 4 \), the inverse process led us to \( f^{-1}(x) = 2(x - 4) \), illustrating how the inverse function reverses the operations in the original function.

Algebraic Manipulation

Algebraic manipulation is a powerful tool that allows us to reshape and solve equations. It's like mathematical origami, where you carefully fold and unfold terms to find a solution. When dealing with inverse functions, algebraic manipulation is key to isolating variables and reversing operations.

• Start by identifying terms that can be moved or changed using basic algebraic operations like addition, subtraction, multiplication, or division.
• Apply these operations systematically to both sides of an equation to maintain balance and derive a desired form, often aiming to solve for a particular variable.

When finding inverses, you use algebraic manipulation to switch the roles of \( x \) and \( y \), effectively reconstructing the equation to achieve \( f^{-1}(x) \).
In our example, we used algebraic manipulation in these key steps:
- Subtract 4 from both sides: \( y - 4 = \frac{1}{2}x \)
- Multiply to eliminate the fraction: \( x = 2(y - 4) \)
Such techniques allow us to dissect functions and reveal their inverses, simplifying complex processes into workable solutions.