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Chapter 9: Problem 62

Graph function and its inverse using the same set of axes. \(g(x)=\frac{1}{4} x+2\)

Short Answer

Expert verified
Graph \(g(x)=\frac{1}{4} x+2\) and \(g^{-1}(x) = 4(x - 2)\) on the same axes.

Step by step solution

01

Understand the Original Function

The given function is \(g(x)=\frac{1}{4} x+2\). This is a linear function with a slope of \(\frac{1}{4}\) and a y-intercept of 2.
02

Plot the Original Function

Plot the points for the function \(g(x)=\frac{1}{4} x+2\). Start by substituting values of x to get corresponding values of g(x). For example, if x = 0, g(x) = 2; if x = 4, g(x) = 3.
03

Find the Inverse Function

To find the inverse function of \(g(x)=\frac{1}{4} x+2\), replace g(x) with y and then solve for x. Start with \(y = \frac{1}{4}x + 2\). Next, isolate x: \(x = 4(y - 2)\). Thus, the inverse function is \(g^{-1}(x) = 4(x - 2)\).
04

Plot the Inverse Function

Plot the points for the inverse function \(g^{-1}(x) = 4(x - 2)\). For example, if x = 2, \g^{-1}(x) = 0\; if x = 3, g^{-1}(x) = 4.
05

Graph Both Functions on the Same Axes

Draw both the original function \(g(x)=\frac{1}{4} x+2\) and its inverse \(g^{-1}(x) = 4(x - 2)\) on the same set of axes. They should be mirror images of each other across the line y = x.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Functions
A linear function is an equation that forms a straight line when graphed. The standard form of a linear function is given as \(y = mx + b\). Here:
  • \(m\) is the slope of the line, indicating its steepness and direction.
  • \(b\) is the y-intercept, the point where the line crosses the y-axis.
In our exercise, the function \(g(x) = \frac{1}{4} x + 2\) is a linear function where the slope \(m = \frac{1}{4}\) and the y-intercept \(b = 2\). This means that for every 4 units we move horizontally (to the right), the line moves 1 unit vertically (upwards). Such an understanding helps in accurately visualizing and plotting the function on a graph.
Inverse Functions
An inverse function essentially 'reverses' the action of the original function. If \(g(x)\) maps \(x\) to \(y\), then \(g^{-1}(x)\) maps \(y\) back to \(x\). To find the inverse, follow these steps:
  • Replace \(g(x)\) with \(y\).
  • Solve the equation for \(x\).
  • Replace \(y\) with \(g^{-1}(x)\).
Using our given function \(g(x) = \frac{1}{4} x + 2\):
  1. Start with \(y = \frac{1}{4} x + 2\).
  2. Isolate \(x\): \[y - 2 = \frac{1}{4} x\]
  3. Multiply by 4: \[x = 4 (y - 2)\]
  4. Thus, the inverse function is \(g^{-1}(x) = 4 (x - 2)\).
The inverse function, when plotted, will reflect the original function across the line \(y = x\), showcasing their symmetry.
Plotting Points
Plotting points is a fundamental skill for graphing functions and their inverses. Here's how to do it effectively:
  • Choose values for \(x\). For example, \(x = 0, 4\) for \(g(x)\) and \(x = 2, 3\) for \(g^{-1}(x)\).
  • Substitute these values into the function to find the corresponding \(y\) values.
  • Plot each \( (x, y) \) pair on the graph鈥檚 coordinate system.
For \(g(x) = \frac{1}{4} x + 2\):
  • If \(x = 0\), then \(g(x) = 2\).
  • If \(x = 4\), then \(g(x) = 3\).
For \(g^{-1}(x) = 4(x - 2)\):
  • If \(x = 2\), then \(g^{-1}(x) = 0\).
  • If \(x = 3\), then \(g^{-1}(x) = 4\).
By plotting these calculated points and connecting them, you can graph both the function and its inverse on the same set of axes, visualizing their relationship easily.

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