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Chapter 1: Problem 56

Graph the function and determine the interval(s) for which \(f(x) \geq 0\). $$f(x)=4 x+2$$

Short Answer

Expert verified
\The intervals for which \(f(x) \geq 0\) are \(f(x) \geq 0\) for all \(x\).

Step by step solution

01

Understanding the linear equation

The function given is \(f(x)=4x+2\), a linear equation in the form \(y=mx+b\), where \(m\) is the slope and \(b\) is the y-intercept. In our case, \(m=4\) and \(b=2\). This means that the graph of the function starts at 2 on the y-axis (when \(x=0\), \(y=2\)), and for every increase by 1 in \(x\), \(y\) increases by \(4\).
02

Graphing the linear function

Start by graphing the y-intercept at (0,2). From that point, the slope \(m=4\) tells us to rise (move upwards) four units and run (move to the right) one unit to draw the line. This represents the graph of the function.
03

Determining where \(f(x) \geq 0\)

To define the region where the function \(f(x) \geq 0\), observe the graph. \(f(x)\) is greater than or equal to zero when the line is on or above the x-axis. With our y-intercept at 2, and the positive slope, we can see that \(f(x)\) is always above the x-axis. Thus, \(f(x) \geq 0\) for all \(x\).

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Most popular questions from this chapter

Determine whether the statement is true or false. Justify your answer. A piecewise-defined function will always have at least one \(x\) -intercept or at least one \(y\) -intercept.

Find a mathematical model that represents the statement. (Determine the constant of proportionality.) \(z\) varies directly as the square of \(x\) and inversely as \(y\) \((z=6 \text { when } x=6 \text { and } y=4 .)\)

Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$f(x)=2.5 x-4.25$$

Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$f(x)=|x-1|$$

(a) Write the linear function \(f\) such that it has the indicated function values and (b) Sketch the graph of the function. $$f\left(\frac{2}{3}\right)=-\frac{15}{2}, \quad f(-4)=-11$$
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