Problem 37 - Algebra for College Students

Question

(a) find \(f^{-1}\) and (b) verify that \(\left(f \circ f^{-1}\right)(x)=x\) and \(\left(f^{-1} \circ f\right)(x)=x\). $$ f(x)=x-4 $$

Step-by-Step Solution

Answer

The inverse is \(f^{-1}(x) = x + 4\). Both compositions equal \(x\).

Step 1: Understand the Function

Given the function is \( f(x) = x - 4 \). This function is linear and one-to-one.

Step 2: Find the Inverse Function

To find the inverse of a function, swap the roles of \( x \) and \( y \). Start with the equation \( y = x - 4 \) and solve for \( x \). Swapping the variables gives \( x = y - 4 \), then solve for \( y \): \( y = x + 4 \). Thus, the inverse function is \( f^{-1}(x) = x + 4 \).

Step 3: Verify the Composition \(\left(f \circ f^{-1}\right)(x) = x\)

Substitute \( f^{-1}(x) = x + 4 \) into \( f(x) \). This gives: \[ (f \circ f^{-1})(x) = f(x + 4) = (x + 4) - 4 = x \].

Step 4: Verify the Composition \(\left(f^{-1} \circ f\right)(x) = x\)

Substitute \( f(x) = x - 4 \) into \( f^{-1}(x) \). This gives: \[ (f^{-1} \circ f)(x) = f^{-1}(x - 4) = (x - 4) + 4 = x \].

Key Concepts

Function Composition

Function composition is the process of combining two functions, say \( f \) and \( g \), to form a new function. This is denoted as \( (f \circ g)(x) \), which means you first apply the function \( g \) to \( x \), and then apply the function \( f \) to the result. In practical terms, it's like taking an input, passing it through the first function, and then passing the output of that function through the second function.
Here's how it works step-by-step:

• Start with your initial input \( x \).
• Apply the first function \( g \) to obtain \( g(x) \).
• Take \( g(x) \) and apply the function \( f \) to it, giving the final result \( f(g(x)) \).

In our original problem, we verified function compositions to ensure that \( (f \circ f^{-1})(x) = x \) and \( (f^{-1} \circ f)(x) = x \). This shows that each function “undoes" the other, leading back to the original input. This behavior is a hallmark of inverse functions.

Linear Function

A linear function is a type of function that creates a straight line when graphed on a coordinate plane. It can be expressed in the form \( f(x) = mx + b \), where:

• \( m \) is the slope, representing the steepness of the line.
• \( b \) is the y-intercept, where the line crosses the y-axis.

In our example, the function \( f(x) = x - 4 \) is linear because it follows the form of a linear equation, though with a slope of 1 and a y-intercept of -4.
Linear functions are straightforward because any change in \( x \) results in a consistent change in \( f(x) \). This simplicity makes them easy to work with, particularly when finding inverse functions. For instance, our function's inverse \( f^{-1}(x) = x + 4 \) is also linear, demonstrating the ease of moving between the original and its inverse.

One-to-One Function

A one-to-one function is a function where each output value is connected to exactly one input value, and vice versa. This property implies that each input maps to a unique output, and each output is matched by a single input. This is crucial for determining the inverse of a function. If a function is not one-to-one, it will not have an inverse that is also a function.
Some key points about one-to-one functions:

• They pass the horizontal line test. When graphed, no horizontal line intersects the function more than once.
• They are often increasing or decreasing throughout their domain.

For our function \( f(x) = x - 4 \), it is one-to-one because it has a consistent slope, which means it either continuously rises or falls, ensuring each output value corresponds to one distinct input. This characteristic is what allows us to find its inverse, \( f^{-1}(x) = x + 4 \), confirming a reversible process between the function and its inverse.