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Chapter 1: Problem 29

Graph the functions \(f, g,\) and \(f+g\) on the same set of coordinate axes. $$ f(x)=\frac{1}{2} x, \quad g(x)=x-1 $$

Short Answer

Expert verified
The graphs of the functions \(f(x)=\frac{1}{2}x\), \(g(x)=x-1\) and their sum \(f(x)+g(x)=\frac{3}{2}x-1\) are straight lines with different slopes and y-intercepts. \(f(x)\) has slope \(\frac{1}{2}\) and y-intercept 0, \(g(x)\) has slope 1 and y-intercept -1 while the sum \(f(x)+g(x)\) has slope \(\frac{3}{2}\) and y-intercept -1.

Step by step solution

01

Identify the properties of the given functions

Firstly, observe the functions given. \(f(x)=\frac{1}{2}x\) is a linear function with slope \(\frac{1}{2}\) and y-intercept 0. \(g(x)=x-1\) is also a linear function with slope 1 and y-intercept -1. Since they are both linear functions, their graphs will be straight lines.
02

Compute the function \(f(x)+g(x)\)

Before graphing the functions, find the function \(f(x)+g(x)\). The sum is computed as follows: \(f(x)+g(x)=\frac{1}{2}x+(x-1)=\frac{3}{2}x-1\). So \(f(x)+g(x)\) is also a linear function with slope \(\frac{3}{2}\) and y-intercept -1.
03

Graph the functions

In this step, graph all the three functions \(f(x), g(x)\), and \(f(x)+g(x)\) on the same set of coordinate axes. For \(f(x)=\frac{1}{2}x\), the line will pass through the origin (0,0) and will be rising at a moderate slope since the slope is \(\frac{1}{2}\). For \(g(x)=x-1\), the y-intercept is -1 and it rises at a steeper slope than \(f(x)\) since its slope is 1. Lastly, graph \(f(x)+g(x)=\frac{3}{2}x-1\). This line has y-intercept -1 and rises at the steepest rate among the three lines, with a slope of \(\frac{3}{2}\).

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