Problem 23 - Precalculus with Limits

Question

In Exercises 23-34, show that \(f\) and \(g\) are inverse functions (a) algebraically and (b) graphically. \(f(x) = 2x\), \(g(x) = \frac{x}{2}\)

Step-by-Step Solution

Answer

Both the algebraic and graphical methods show that the functions \(f(x) = 2x\) and \(g(x) = \frac{x}{2}\) are inverses of each other.

Step 1: Algebraic Method

To show inverse algebraically, the substitution method is used. Substitute \(y = f(x) = 2x\) into \(g(y)\) and solve. If \(g(f(x)) = x\), then the functions are inverses. Thus, \(g(f(x)) = g(2x) = \frac{2x}{2} = x\). Therefore, using the algebraic method, \(f(x)\) and \(g(x)\) are inverses.

Step 2: Graphical Method

To show inverses graphically, the graphs of the functions need to be reflections of each over the line \(y = x\). The graph of \(f(x) = 2x\) is a straight line passing through the origin and the slope is 2. The graph of \(g(x) = \frac{x}{2}\) is also a straight line passing through the origin but with slope \(\frac{1}{2}\). On reflection of \(f(x) = 2x\) in the line \(y = x\), we get the line \(g(x) = \frac{x}{2}\). Therefore, using the graphical method, \(f(x)\) and \(g(x)\) are inverses.

Key Concepts

Algebraic Method for Inverses

Understanding the algebraic method for finding inverses can streamline the process of verifying if two functions are indeed inverse pairs.

Let's consider the provided exercise where we have two functions, \( f(x) = 2x \) and \( g(x) = \frac{x}{2} \). To prove algebraically that they are inverses, we apply the substitution method. This involves replacing the variable x with the expression for \( f(x) \) in the second function \( g(x) \). If the result simplifies to x, then we deem them inverses because applying \( g \) after \( f \) brings us back to our original input.

In this case, after substituting, we get \( g(f(x)) = g(2x) = \frac{2x}{2} \) which simplifies to x. Therefore, the algebraic manipulation confirms that \( f \) and \( g \) are inverse functions.

Graphical Method for Inverses

Visually determining inverses can be as effective as the algebraic method and sometimes even more intuitive. To use the graphical method, one must understand that the graphs of inverse functions are reflections of each other over the line \( y = x \).

Looking at the exercise, \( f(x) = 2x \) has a graph which is a straight line with a slope of 2, and \( g(x) = \frac{x}{2} \) has a graph that is also a straight line but with a slope of \( \frac{1}{2} \). To check for inversedness, we reflect the graph of \( f(x) \) over the line \( y = x \), and if it aligns with the graph of \( g(x) \) , then they are graphical inverses.

In this exercise, this reflection gives us the line corresponding to \( g(x) \), confirming that they are inverses. This graphical method not only provides a visual confirmation but also reinforces the concept of inverse functions being mirror images across the line \( y = x \) .

Reflection Over the Line y = x

The concept of reflection over the line \( y = x \) plays a crucial role in understanding inverse functions. This specific line serves as a mirror, indicating a perfect switch between x and y coordinates.

For any point \( (a, b) \) on the graph of a function, its inverse will have the point \( (b, a) \) after reflecting over the line \( y = x \). This is because, in inverse functions, inputs and outputs exchange roles. Therefore, if a function takes an input of a and produces an output of b, its inverse will take an input of b to produce an output of a.

In practical terms, to reflect a graph across the line \( y = x \) , every point \( (x, y) \) on the original graph is swapped to \( (y, x) \) on the reflected graph. This geometric interpretation reinforces the understanding of the reversal of roles in inverse functions and provides a way to graphically verify inversedness.