Problem 36 - Precalculus : Building Concepts and Connections

Question

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$g(x)=\frac{4}{3} x$$

Step-by-Step Solution

Answer

The inverse of the function \(g(x)= \frac{4}{3}x\) is \(g^{-1}(x) = \frac{3}{4}x\).

Step 1: Identify the Given Function

First, identify the function that is given. In this case, the function is \( g(x)= \frac{4}{3}x\).

Step 2: Find the Inverse

To find the inverse of a function, replace the \(g(x)\) with \(y\) to get \(y = \frac{4}{3}x \). Then, switch the roles of \(x\) and \(y\) to get \(x = \frac{4}{3}y \). In the last step, solve for \(y\) which gives us the inverse function. Dividing both sides of the equation by \( \frac{4}{3} \) yields \(y = \frac{3}{4}x\). Thus, the inverse of the function \( g(x) \) is \( g^{-1}(x) = \frac{3}{4}x\).

Step 3: Graph the Functions

Graph both functions, \(g(x) = \frac{4}{3}x \) and \(g^{-1}(x) = \frac{3}{4}x\), on the same set of axes. Both functions are linear, and their graphs will be straight lines. The line \( g(x) = \frac{4}{3}x \) has a slope of \( \frac{4}{3} \) and passes through the origin. The line \( g^{-1}(x) = \frac{3}{4}x \) also passes through the origin but has a slope of \( \frac{3}{4} \). The functions are reflections of each other across the line \(y=x\).

Key Concepts

Linear Functions

Linear functions are one of the simplest types of functions in mathematics. They follow the form \(f(x) = mx + b\), where \(m\) represents the slope of the line, and \(b\) is the y-intercept, which is the point where the line crosses the y-axis. A linear function creates a straight line on a graph, hence the name.

In the exercise, the function given is \(g(x) = \frac{4}{3}x\). Notice how it matches the form of a linear function with a slope \(m = \frac{4}{3}\) and a y-intercept of 0. This means the line passes through the origin (0,0). The slope tells us how steep the line is and in which direction it goes:

• A positive slope indicates the line rises as it moves from left to right.
• A negative slope indicates the line falls as it moves from left to right.

Understanding the role of slope and intercept allows us to visualize and predict the behavior of linear functions effectively, making them a fundamental concept in graphing and algebra.

Graphing Functions

Graphing functions provides a visual representation of the relationship between variables. When graphing the function \(g(x) = \frac{4}{3}x\), we begin at the y-intercept, which is 0 in this case, and use the slope to determine the direction of the line. For each step horizontally to the right, you move up \(\frac{4}{3}\) units vertically.

To graph the inverse, \(g^{-1}(x) = \frac{3}{4}x\), start again at the origin. Move 4 units to the right and upward 3 units for each point along the line. Both lines should intersect at the origin and be reflections across the line \(y=x\).

• The process of graphing helps in understanding how the original function and its inverse relate to each other.
• Graphing can also help verify that computations of inverses are correct, as they should mirror about the line \(y=x\).

Properly graphing functions and their inverses is a powerful tool that gives us insights into mathematical relationships.

Function Inverses

Finding the inverse of a function can reveal essential insights into the nature of mathematical operations. The inverse function essentially "undoes" the effects of the original function. To find an inverse, swap the \(x\) and \(y\) in the equation and solve for \(y\).

In this exercise, the function \(g(x) = \frac{4}{3}x\) was inverted by switching the roles of \(x\) and \(y\), resulting in \(x = \frac{4}{3}y\). Solving for \(y\) gives the inverse \(g^{-1}(x) = \frac{3}{4}x\). This inverse function allows us to reverse the effect of \(g(x)\) on any input.

• Function inverses often give us a different perspective on the mathematical operations being performed.
• When graphed, a function and its inverse are reflected across the line \(y=x\), highlighting their symmetry.

Understanding inverses is crucial in various fields, including solving equations in algebra, and is fundamental in calculus on concepts like finding derivatives and integrals.