Explain why or why not. Determine whether the following

Piecewise linear functions The parking costs in a city garage are $2.00 for the first half hour and $1.00 for each additional half hour. Graph the function C = f(t) that gives the cost of parking for t hours, where \(0 \leq t \leq 3\).

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. A function could have the property that f(x) = f(-x) for all x. b. cos (a + b) = cos a + cos b for all a and b in \([0,\ 2\pi]\). c. If f is a linear function of the form f(x) = mx + b, then f(u + v) = f(u) + f(v) for all u and v. d. The function f(x) = 1 - x has the property that f(f(x)) = x. e. The set {x: |x + 3| > 4} can be drawn on the number line without lifting your pencil. f. \(\log _{10}(x y)=\left(\log _{10} x\right)\left(\log _{10} y\right)\) g. \(\sin ^{-1}(\sin (2 \pi))=0\)

Function from words Suppose you plan to take a 500-mi trip in a car that gets 35 mi/gal. Find the function C = f(p) that gives the cost of gasoline for the trip when gasoline costs $p per gallon.

Domain and range Find the domain and range of the following functions. a. \(f(x)=x^{5}+\sqrt{x}\) b. \(g(y)=\frac{1}{y-2}\) c. \(h(z)=\sqrt{z^{2}-2 z-3}\)

Equations of lines Find an equation of the lines with the following properties. Graph the lines. a. The line passing through the points (2, -3) and (4, 2) b. The line with slope \(\frac{3}{4}\) and x-intercept (-4, 0) c. The line with intercepts (4, 0) and (0, -2)

Graphing absolute value Consider the function f(x) = 2 (x - | x | ). Express the function in two pieces without using the absolute value. Then graph the function by hand. Use a graphing utility only to check your work.

Graphing equations Graph the following equations. Use a graphing utility only to check your work. a. 2x - 3y + 10 = 0 b. \(y=x^{2}+2 x-3\) c. \(x^{2}+2 x+y^{2}+4 y+1=0\) d. \(x^{2}-2 x+y^{2}-8 y+5=0\)

Root functions Graph the function \(f(x)=x^{1 / 3}\) and \(g(x)=x^{1 / 4}\). Find all points where the two graphs intersect. For x > 1, is f(x) > g(x) or is g(x) > f(x)?

Root functions Find the domain and range of the functions \(f(x)=x^{1 / 7}\) and \(g(x)=x^{1 / 4}\).

Intersection points Graph the equations \(y=x^{2}\) and \(x^{2}+y^{2}-7 y+8=0\). At what point(s) do the curves intersect?

Boiling-point function Water boils at \(212^{\circ}\mathrm{\ F}\) at sea level and at \(180^{\circ}\mathrm{\ F}\) at an elevation of 6000 ft. Assume that the boiling point B varies linearly with altitude a. Find the function B = f(a) that describes the dependence. Comment on whether a linear function gives a realistic model.

Publishing costs A small publisher plans to spend $1000 for advertising a paperback book and estimates the printing cost is $2.50 per book. The publisher will receive $7 for each book sold. a. Find the function C = f(x) that gives the cost of producing x books. b. Find the function R = g(x) that gives the revenue from selling x books. c. Graph the cost and revenue functions and find the number of books that must be sold for the publisher to break even.

Shifting and scaling The graph of y = f(x) is shown in the figure. Sketch the graph of the following functions. a. f(x + 1) b. 2f(x - 1) c. -f(x/2) d. f(2(x - 1))

Composite functions Let \(f(x)=x^3,\ g(x)=\sin x\) and \(h(x)=\sqrt{x}\) a. Evaluate \(h(g(\pi / 2))\). b. Find h(f(x)). c. Find f(g(h(x))). d. Find the domain of \(g\ \circ\ f\). e. Find the range of \(f\ \circ\ g\).

Symmetry Identify the symmetry in the graphs of the following equations. a. y = cos 3x b. \(y=3 x^{4}-3 x^{2}+1\) c. \(y^{2}-4 x^{2}=4\)

18-19. Properties of logarithms and exponentials Use properties of logarithms and exponentials, not a calculator, for the following exercises. Solve the equation \(48=6 e^{4 k}\) for k.

18-19. Properties of logarithms and exponentials Use properties of logarithms and exponentials, not a calculator, for the following exercises. Solve the equation \(\log x^{2}+3 \log x=\log 32\) for x. Does the answer depend on the base of the log?

Graphs of logarithmic and exponential functions The figure shows the graphs of \(y=2^x,\ y=3^{-x}\), and y = -In x. Match each curve with the correct function.

21-22. Existence of inverses Use analytical methods and/or graphing to determine the intervals on which the following functions have an inverse. \(f(x)=x^{3}-3 x^{2}\)

Use analytical methods and/or graphing to determine the intervals on which the following functions have an inverse. g(t) = 2 sin (t/3)

Find the inverse on the specified interval and express it in the form \(y=f^{-1}(x)\). Then graph f and \(f^{-1}\). \(f(x)=x^{2}-4 x+5 \text { for } x>2\)

Find the inverse on the specified interval and express it in the form \(y=f^{-1}(x)\). Then graph f and \(f^{-1}\). \(f(x)=1 / x^{2} \text { for } x>0\)

Degrees and radians: a. What is the radian measure of a \(135^{\circ}\) angle? b. What is the degree measure of a \(4 \pi / 5 \text {-rad }\) angle? c. What is the length of the arc of a circle of radius 10 associated with an angle of \(4 \pi / 3 \quad \mathrm{rad}\)?

Find a trigonometric function f that satisfies each set of properties. Answers are not unique. a. It has a period of 6 hr with a minimum value of -2 at t = 0 hr and a maximum value of 2 at t = 3 hr. b. It has a period of 24 hr with a maximum value of 20 at t = 6 hr and a minimum value of 10 at t = 18 hr.

Without using a calculator, evaluate or simplify the following expressions. \(\sin ^{-1}(\sqrt{3} / 2)\)

Problem 29RE Graph to function: Find a trigonometric func?tion ? represented by the graph in the figure.

Without using a calculator, evaluate or simplify the following expressions. \(\cos ^{-1}(\sqrt{3} / 2)\)

Without using a calculator, evaluate or simplify the following expressions. \(\cos ^{-1}\left(-\frac{1}{2}\right)\)

Without using a calculator, evaluate or simplify the following expressions. \(\sin ^{-1}(-1)\)

Without using a calculator, evaluate or simplify the following expressions. \(\cos \left(\cos ^{-1}(-1)\right)\)

Without using a calculator, evaluate or simplify the following expressions. \(\sin \left(\sin ^{-1} x\right)\)

Without using a calculator, evaluate or simplify the following expressions. \(\cos ^{-1}(\sin 3 \pi)\)

Given that \(\theta=\sin ^{-1}\left(\frac{12}{13}\right)\), evaluate \(\cos \theta\), \(\tan \theta\), \(\cot \theta\), \(\sec \theta\), and \(\csc \theta\).

Draw a right triangle to simplify the given expression Assume x > 0 and \(0 \leq \theta \leq \pi / 2\). \(\cos \left(\tan ^{-1} x\right)\)

Draw a right triangle to simplify the given expression Assume x > 0 and \(0 \leq \theta \leq \pi / 2\). \(\sin \left(\cos ^{-1}(x / 2)\right)\)

Draw a right triangle to simplify the given expression Assume x > 0 and \(0 \leq \theta \leq \pi / 2\). \(\tan \left(\sec ^{-1}(x / 2)\right)\)

Draw a right triangle to simplify the given expression Assume x > 0 and \(0 \leq \theta \leq \pi / 2\). \(\cot ^{-1}(\tan \theta)\)

Draw a right triangle to simplify the given expression Assume x > 0 and \(0 \leq \theta \leq \pi / 2\). \(\csc ^{-1}(\sec \theta)\)

Draw a right triangle to simplify the given expression Assume x > 0 and \(0 \leq \theta \leq \pi / 2\). \(\sin \left(2 \cos ^{-1} x\right)\) (Hint: Use \(\sin 2 \theta=2 \sin \theta \cos \theta\).)

Draw a right triangle to simplify the given expression Assume x > 0 and \(0 \leq \theta \leq \pi / 2\). \(\cos \left(2 \sin ^{-1} x\right)\) (Hint: Use \(\cos 2 \theta=\cos ^{2} \theta-\sin ^{2} \theta\).)

A common way of displaying a sphere (such as Earth) on a plane (such as a map) is to use a stereographic projection. Here is the two-dimensional version of the method, which maps a circle to a line. Let P be a point on the right half of a circle of radius R identified by the angle \(\varphi\). Find the function \(x=F(\varphi)\) that gives the x-coordinate \((x \geq 0)\) corresponding to \(\varphi\) for \(0<\varphi \leq \pi\).

Problem 63PP Growth Speed The images of trees in Figure P1.68 come from a catalog advertising fast-growing trees. If we mark the position of the top of the tree in the successive years, as shown in the graph in the figure, we obtain a motion diagram much like ones we have seen for other kinds of motion. The motion isn’t steady, of course. In some months the tree grows rapidly; in other months, quite slowly. We can see, though, that the average speed of growth is fairly constant for the first few years. What is the tree’s speed of grow?th, in feet ?per year, from ?t = 1 yr t? = 3 yr? A. 12 ft/yr B. 9 ft/yr C. 6 ft/yr D. 3 ft/yr

Problem 64PP Growth Speed The images of trees in Figure P1.68 come from a catalog advertising fast-growing trees. If we mark the position of the top of the tree in the successive years, as shown in the graph in the figure, we obtain a motion diagram much like ones we have seen for other kinds of motion. The motion isn’t steady, of course. In some months the tree grows rapidly; in other months, quite slowly. We can see, though, that the average speed of growth is fairly constant for the first few years. What is this speed in m/s?

Shifting and scaling Starting with the graph of \(f(x)=x^{2}\), sketch the graph of the following functions. Use a graphing calculator only to check your work. a. f(x + 3) b. 2f(x - 4) c. -f(3x) d. f(2x - 3))

Composite functions Find functions f and g such that \(h=f\ \circ\ g\) a. \(h(x)=\sin \left(x^{2}+1\right)\) b. \(h(x)=\left(x^{2}-4\right)^{-3}\) c. \(h(x)=e^{\cos 2 x}\)

Use shifts and scalings to graph the following functions, and identify the amplitude and period. a. f(x)=4 cos (x/2) b. \(g(\theta)=2 \sin (2 \pi \theta / 3)\) c. \(h(\theta)=-\cos (2(\theta-\pi / 4))\)

Graph to function Find a trigonometric function f represented by the graph in the figure.

Draw a right triangle to simplify the given expression Assume x > 0 and \(0 \leq \theta \leq \pi / 2\). \(\sin ^{-1} x+\sin ^{-1}(-x)\)