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Chapter 12: Problem 33

Graph the function as a solid line (or curve) and then graph its inverse on the same set of axes as a dashed line (or curve). \(g(x)=-4 x\)

Short Answer

Expert verified
The function is \( g(x) = -4x \) and its inverse is \( g^{-1}(x) = -\frac{x}{4} \). Plot both and verify symmetry about \( y = x \).

Step by step solution

01

Understand the Function

The given function is a linear function represented by the equation: \( g(x) = -4x \).
02

Plot the Given Function

To plot the function \( g(x) = -4x \), create a table of values for \( x \) and \( g(x) \). For example:\[ \begin{array}{c|c} x & g(x) \hline -2 & 8 \ -1 & 4 \ 0 & 0 \ 1 & -4 \ 2 & -8 \end{array}\]Plot these points on a graph and connect them with a solid line.
03

Find the Inverse Function

The inverse function of \( g(x) = -4x \) is found by swapping \( x \) and \( y \) and solving for \( y \). Start with \( y = -4x \), then replace \( y \) with \( x \) and \( x \) with \( y \):\[ x = -4y \].Solving for \( y \) gives \( y = -\frac{x}{4} \). So the inverse function is \( g^{-1}(x) = -\frac{x}{4} \).
04

Plot the Inverse Function

To plot the inverse function \( g^{-1}(x) = -\frac{x}{4} \), create a table of values for \( x \) and \( g^{-1}(x) \). For example:\[ \begin{array}{c|c} x & g^{-1}(x) \hline -8 & 2 \ -4 & 1 \ 0 & 0 \ 4 & -1 \ 8 & -2 \end{array}\]Plot these points on the same graph and connect them with a dashed line.
05

Verify Symmetry

Ensure that the graph of the function and its inverse are symmetrical about the line \( y = x \). This verifies that they are indeed inverses of each other.

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Key Concepts

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

Linear Functions
Linear functions produce a straight line when graphed. These functions have the general form: y = mx + c, where
  • y is the dependent variable
  • x is the independent variable
  • m is the slope of the line
  • c is the y-intercept
For the function g(x) = -4x, m = -4 and c = 0. This tells us that the slope is -4 and it crosses the y-axis at (0,0). The negative slope indicates that the line decreases from left to right. To better understand, you can choose specific values for x, calculate their corresponding y-values and plot these points to visualize the linear function.
Inverse Functions
Inverse functions reverse the roles of the dependent and independent variables. If a function is represented as f(x) and it maps x to y, the inverse function maps y back to x. To find the inverse of a function g(x) = -4x, we follow these steps:
  • Rewrite the function using y: y = -4x
  • Switch x and y: x = -4y
  • Solve for y: y = -x / 4
Thus, the inverse function is g^{-1}(x) = -x / 4. When plotting, the points switch places, vertically reflecting across the line y = x.
Plotting Graphs
Plotting graphs is crucial to understand functions visually. Start with a table of values for your function so you can determine points to plot. For g(x) = -4x, select x-values (e.g., -2, -1, 0, 1, 2) and compute corresponding y-values. Mark these points on the graph, and connect them with a solid line. Repeat this process for the inverse function g^{-1}(x) = -x / 4 using selected x-values (e.g., -8, -4, 0, 4, 8). Connect these points with a dashed line. This helps illustrate the changes between the function and its inverse.
Function Symmetry
Graphs of a function and its inverse exhibit symmetry about the line y = x. This means that if you fold the graph along the line y = x, the function and its inverse will align perfectly. By plotting g(x) = -4x and its inverse g^{-1}(x) = -x / 4, you can visually confirm their symmetry. This relationship verifies the inverses, ensuring they reflect accurately across y = x. Checking for symmetry is an effective way to confirm that you have correctly identified and plotted inverse functions.

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