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Chapter 1: Problem 3
Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$f(x)=3 x+8 \text { and } g(x)=\frac{x-8}{3}$$
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Furry Finances. A pet insurance policy has a monthly rate that is a function of the age of the insured dog or cat. For pets whose age does not exceed \(4,\) the monthly cost is \(\$ 20 .\) The cost then increases by \(\$ 2\) for each successive year of the pet's age. $$\begin{array}{cc} \text { Age Not Exceeding } & \text { Monthly cost } \\ \hline 4 & \$ 20 \\ 5 & \$ 22 \\ 6 & \$ 24 \end{array}$$ The cost schedule continues in this manner for ages not exceeding \(10 .\) The cost for pets whose ages exceed 10 is \(\$ 40 .\) Use this information to create a graph that shows the monthly cost of the insurance, \(f(x),\) for a pet of age \(x,\) where the function's domain is [0,14]
A rectangular coordinate system with coordinates in miles is placed with the origin at the center of Los Angeles. The figure indicates that the University of Southern California is located 2.4 miles west and 2.7 miles south of central Los Angeles. A seismograph on the campus shows that a small earthquake occurred. The quake's epicenter is estimated to be approximately 30 miles from the university. Write the standard form of the equation for the set of points that could be the epicenter of the quake.
a. Graph the functions \(f(x)=x^{n}\) for \(n=2,4,\) and 6 in a [-2,2,1] by [-1,3,1] viewing rectangle. b. Graph the functions \(f(x)=x^{n}\) for \(n=1,3,\) and 5 in a [-2,2,1] by [-2,2,1] viewing rectangle. c. If \(n\) is positive and even, where is the graph of \(f(x)=x^{n}\) increasing and where is it decreasing? d. If \(n\) is positive and odd, what can you conclude about the graph of \(f(x)=x^{n}\) in terms of increasing or decreasing behavior? e. Graph all six functions in a [-1,3,1] by [-1,3,1] viewing rectangle. What do you observe about the graphs in terms of how flat or how steep they are?
Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$for the given function. $$f(x)=\frac{1}{x}$$
Sketch the graph of \(f\) using the following properties. (More than one correct graph is possible.) \(f\) is a piecewise function that is decreasing on \((-\infty, 2), f(2)=0, f\) is increasing on \((2, \infty),\) and the range of \(f\) is \([0, \infty)\)
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