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Chapter 1: Problem 32

Graph the functions \(f, g,\) and \(f+g\) on the same set of coordinate axes. $$ f(x)=4-x^{2}, \quad g(x)=x $$

Short Answer

Expert verified
Graph \(f(x)\) as a downward-facing parabola with the vertex at (0, 4), graph \(g(x)\) as a line with a slope 1, and graph \(f(x) + g(x)\) by adding the corresponding \(y\) values from \(f(x)\) and \(g(x)\) for each \(x\). Then superimpose all three graphs onto the same axes.

Step by step solution

01

Graph function \(f(x)\)

First graph \(f(x) = 4 - x^2\). This is a downward-facing parabola with the vertex at (0, 4), meaning it peaks at 4 and gets smaller as \(x\) moves away from 0 on either side.
02

Graph function \(g(x)\)

Next graph \(g(x) = x\). This is a straight line with slope 1, meaning for every step to the right, the line goes one step up. It passes through the origin.
03

Graph function \(f(x) + g(x)\)

Finally, graph \(f(x) + g(x)\), which implies that for every \(x\), take the corresponding \(y\) values on both \(f(x)\) and \(g(x)\) and add them. Those sums are the \(y\) coordinates for the corresponding \(x\) coordinates on the \(f(x)+g(x)\) curve.
04

Superimpose all three functions

Now superimpose the graphs of \(f(x)\), \(g(x)\), and \(f(x) + g(x)\) onto the same set of coordinate axes. This will show the relationship between the functions and how adding \(f\) and \(g\) creates the third graph.

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