In Exercises 67 and 68, Identify functions for and (There are many correct answers.)F x

In Exercises 1 and 2, use the graphs of f and g to answer the following. (a) Identify the domains and ranges of f and g. (b) Identify f(-2) and g(3). (c) For what value(s) of x is f(x) = g(x)? (d) Estimate the solution(s) of f(x) = 2. (e) Estimate the solutions of g(x) = 0.

In Exercises 1 and 2, use the graphs of f and g to answer the following. (a) Identify the domains and ranges of f and g. (b) Identify f(-2) and g(3). (c) For what value(s) of x is f(x) = g(x)? (d) Estimate the solution(s) of f(x) = 2. (e) Estimate the solutions of g(x) = 0.

In Exercises 3-12, evaluate (if possible) the function at the given value(s) of the independent variable. Simplify the results. f(x) = 7x - 4 (a) f(0) (b) f(-3) (c) f(b) (d) f(x - 1)

In Exercises 3-12, evaluate (if possible) the function at the given value(s) of the independent variable. Simplify the results. \(f(x)=\sqrt{x+5}\) (a) f(-4) (b) f(11) (c) f(-8) (d) \(f(x+\Delta x)\) Text Transcription: f(x) = sqrt x+5 f(x + Delta x)

In Exercises 3-12, evaluate (if possible) the function at the given value(s) of the independent variable. Simplify the results. \(g(x)=5-x^{2}\) (a) g(0) (b) \(g(\sqrt{5})\) (c) g(-2) (d) g(t - 1) Text Transcription: g(x) = 5 - x^2 g(sqrt 5)

In Exercises 3-12, evaluate (if possible) the function at the given value(s) of the independent variable. Simplify the results. \(g(x)=x^{2}(x-4)\) (a) g(4) (b) \(g\left(\frac{3}{2}\right)\) (c) g(c) (d) g(t + 4) Text Transcription: g(x) = x^2(x - 4) g(3/2)

In Exercises 3-12, evaluate (if possible) the function at the given value(s) of the independent variable. Simplify the results. f(x) = cos 2x (a) f(0) (b) \(f(-\pi / 4)\) (c) \(f(\pi / 3)\) Text Transcription: f(-pi/4) f(pi/3)

In Exercises 3-12, evaluate (if possible) the function at the given value(s) of the independent variable. Simplify the results. f(x) = sin x (a) \(f(\pi)\) (b) \(f(5 \pi / 4)\) (c) \(f(2 \pi / 3)\) Text Transcription: f(pi) f(5pi/4) f(2pi/3)

In Exercises 3-12, evaluate (if possible) the function at the given value(s) of the independent variable. Simplify the results. \(f(x)=x^{3}\) \(\frac{f(x+\Delta x)-f(x)}{\Delta x}\) Text Transcription: f(x) = x^3 f(x + Delta x) - f(x) / Delta x

In Exercises 3-12, evaluate (if possible) the function at the given value(s) of the independent variable. Simplify the results. f(x) = 3x - 1 \(\frac{f(x)-f(1)}{x-1}\) Text Transcription: f(x) - f(1) / x - 1

In Exercises 3-12, evaluate (if possible) the function at the given value(s) of the independent variable. Simplify the results. \(f(x)=\frac{1}{\sqrt{x-1}}\) \(\frac{f(x)-f(2)}{x-2}\) Text Transcription: f(x) = 1 / sqrt x-1 f(x) - f(2) / x - 2

In Exercises 3-12, evaluate (if possible) the function at the given value(s) of the independent variable. Simplify the results. \(f(x)=x^{3}-x\) \(\frac{f(x)-f(1)}{x-1}\) Text Transcription: f(x) = x^2 - x f(x) - f(1) / x - 1

In Exercises 13-20, find the domain and range of the function. \(f(x)=4 x^{2}\) Text Transcription: f(x) = 4x^2

In Exercises 13-20, find the domain and range of the function. \(g(x)=x^{2}-5\) Text Transcription: g(x) = x^2 - 5

In Exercises 13-20, find the domain and range of the function. \(g(x)=\sqrt{6 x}\) Text Transcription: g(x) = sqrt 6x

In Exercises 13-20, find the domain and range of the function. \(h(x)=-\sqrt{x+3}\) Transcription: h(x) = - sqrt x+3

In Exercises 13-20, find the domain and range of the function. \(f(t)=\sec \frac{\pi t}{4}\) Text Transcription: f(t) = sec pi t / 4

In Exercises 13-20, find the domain and range of the function. h(t) = cot t

In Exercises 13-20, find the domain and range of the function. \(f(x)=\frac{3}{x}\) Text Transcription: f(x) = 3/x

In Exercises 13-20, find the domain and range of the function. \(g(x)=\frac{2}{x-1}\) Text Transcription: g(x) = 2/x-1

In Exercises 21-26, find the domain of the function. \(f(x)=\sqrt{x}+\sqrt{1-x}\) Text Transcription: f(x) = sqrt x + sqrt 1-x

In Exercises 21-26, find the domain of the function. \(f(x)=\sqrt{x^{2}-3 x+2}\) Text Transcription: f(x) = sqrt x^2 - 3x + 2

In Exercises 21-26, find the domain of the function. \(g(x)=\frac{2}{1-\cos x}\) Text Transcription: g(x) = 2 / 1 - cos x

In Exercises 21-26, find the domain of the function. \(h(x)=\frac{1}{\sin x-\frac{1}{2}}\) Text Transcription: h(x) = 1 / sin x - 1/2

In Exercises 21-26, find the domain of the function. \(f(x)=\frac{1}{|x+3|}\) Text Transcription: f(x) = 1 / |x+3|

In Exercises 21-26, find the domain of the function. \(g(x)=\frac{1}{\left|x^{2}-4\right|}\) Text Transcription: g(x) = 1 / |x^2 - 4|

In Exercises 27-30, evaluate the function as indicated. Determine its domain and range. \(f(x)=\left\{\begin{array}{ll} 2 x+1, & x<0 \\ 2 x+2, & x \geq 0 \end{array}\right.\) (a) f(-2) (b) f(0) (c) f(1) (d) \(f\left(t^{2}+1\right)\) Text Transcription: f(x) = { _2x + 2, x geq 0 ^2x + 1, x < 0 f(t^2 + 1)

In Exercises 27-30, evaluate the function as indicated. Determine its domain and range. \(f(x)=\left\{\begin{array}{ll} x^{2}+2, & x \leq 1 \\ 2 x^{2}+2, & x>1 \end{array}\right.\) (a) f(-2) (b) f(0) (c) f(1) (d) \(f\left(s^{2}+2\right)\) Text Transcription: f(x) = { _2x^2 + 2, x > 1 ^x^2 + 2, x leq 1 f(s^2 + 2)

In Exercises 27-30, evaluate the function as indicated. Determine its domain and range. \(f(x)=\left\{\begin{array}{ll} |x|+1, & x<1 \\ -x+1, & x \geq 1 \end{array}\right.\) (a) f(-3) (b) f(1) (c) f(3) (d) \(f\left(b^{2}+1\right)\) Text Transcription: f(x) = { _-x + 1, x geq 1 ^|x| + 1, x < 1 f(b^2 + 1)

In Exercises 27-30, evaluate the function as indicated. Determine its domain and range. \(f(x)=\left\{\begin{array}{l} \sqrt{x+4}, x \leq 5 \\ (x-5)^{2}, x>5 \end{array}\right.\) (a) f(-3) (b) f(0) (c) f(5) (d)f(10) Text Transcription: f(x) = { _(x - 5)^2, x > 5 ^ sqrt x+4, x leq 5

In Exercises 31-38, sketch a graph of the function and find its domain and range. Use a graphing utility to verify your graph. f(x) = 4 - x

In Exercises 31-38, sketch a graph of the function and find its domain and range. Use a graphing utility to verify your graph. \(g(x)=\frac{4}{x}\) Text Transcription: g(x) = 4/x

In Exercises 31-38, sketch a graph of the function and find its domain and range. Use a graphing utility to verify your graph. \(h(x)=\sqrt{x-6}\) Text Transcription: h(x) = sqrt x-6

In Exercises 31-38, sketch a graph of the function and find its domain and range. Use a graphing utility to verify your graph. \(f(x)=\frac{1}{4} x^{3}+3\) Text Transcription: f(x) = 1/4 x^3 + 3

In Exercises 31-38, sketch a graph of the function and find its domain and range. Use a graphing utility to verify your graph. \(f(x)=\sqrt{9-x^{2}}\) Text Transcription: f(x) = sqrt 9-x^2

In Exercises 31-38, sketch a graph of the function and find its domain and range. Use a graphing utility to verify your graph. \(f(x)=x+\sqrt{4-x^{2}}\) Text Transcription: f(x) = x + sqrt 4-x^2

In Exercises 31-38, sketch a graph of the function and find its domain and range. Use a graphing utility to verify your graph. \(g(t)=3 \sin \pi t\) Text Transcription: g(t) = 3 sin pi t

In Exercises 31-38, sketch a graph of the function and find its domain and range. Use a graphing utility to verify your graph. \(h(\theta)=-5 \cos \frac{\theta}{2}\) Text Transcription: h(theta) = -5 cos theta/2

The graph of the distance that a student drives in a 10-minute trip to school is shown in the figure. Give a verbal description of characteristics of the student's drive to school.

A student who commutes 27 miles to attend college remembers, after driving a few minutes, that a term paper that is due has been forgotten. Driving faster than usual, the student returns home, picks up the paper, and once again starts toward school. Sketch a possible graph of the student's distance from home as a function of time.

In Exercises 41-44, use the Vertical Line Test to determine whether y is a function of x. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. \(x-y^{2}=0\) Text Transcription: x - y^2 = 0

In Exercises 41-44, use the Vertical Line Test to determine whether y is a function of x. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. \(\sqrt{x^{2}-4}-y=0\) Text Transcription: sqrt x^2 - 4 - y = 0

In Exercises 41-44, use the Vertical Line Test to determine whether y is a function of x. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. \(y=\left\{\begin{aligned} x+1, & \ \ x \leq 0 \\ -x+2, & \ \ x>0 \end{aligned}\right.\) Text Transcription: y = { _-x + 2, x > 0 ^x + 1, x leq 0

In Exercises 41-44, use the Vertical Line Test to determine whether y is a function of x. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. \(x^{2}+y^{2}=4\) Text Transcription: x^2 + y^2 = 4

In Exercises 45-48, determine whether y is a function of x. \(x^{2}+y^{2}=16\) Text Transcription: x^2 + y^2 = 16

In Exercises 45-48, determine whether y is a function of x. \(x^{2}+y=16\) Text Transcription: x^2 + y = 16

In Exercises 45-48, determine whether y is a function of x. \(y^{2}=x^{2}-1\) Text Transcription: y^2 = x^2 - 1

In Exercises 45-48, determine whether y is a function of x. \(x^{2} y-x^{2}+4 y=0\) Text Transcription: x^2 - x^2 + 4y = 0

In Exercises 49-54, use the graph of y = f(x) to match the function with its graph. y = f(x + 5)

In Exercises 49-54, use the graph of y = f(x) to match the function with its graph. y = f(x) - 5

In Exercises 49-54, use the graph of y = f(x) to match the function with its graph. y = -f(-x) - 2

In Exercises 49-54, use the graph of y = f(x) to match the function with its graph. y = -f(x - 4)

In Exercises 49-54, use the graph of y = f(x) to match the function with its graph. y = f(x + 6) + 2

In Exercises 49-54, use the graph of y = f(x) to match the function with its graph. y = f(x - 1) + 3

Use the graph of f shown in the figure to sketch the graph of each function. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. (a) f(x + 3) (b) f(x - 1) (c) f(x) + 2 (d) f(x) - 4 (e) 3f(x) (f) \(\frac{1}{4} f(x)\) Text Transcription: 1/4 f(x)

Use the graph of f shown in the figure to sketch the graph of each function. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. (a) f(x - 4) (b) f(x + 2) (c) f(x) + 4 (d) f(x) - 1 (e) 2f(x) (f) \(\frac{1}{2} f(x)\) Text Transcription: 1/2 f(x)

57. Use the graph of \(f(x)=\sqrt{x}\) to sketch the graph of each function. In each case, describe the transformation. (a) \(y=\sqrt{x}+2\) (b) \(y=-\sqrt{x}\) (c) \(y=\sqrt{x-2}\) Text Transcription: f(x) = sqrt x y = sqrt x + 2 y = - sqrt x y = sqrt x-2

Specify a sequence of transformations that will yield each graph of h from the graph of the function f(x) = sin x. (a) \(h(x)=\sin \left(x+\frac{\pi}{2}\right)+1\) (b) h(x) = -sin(x - 1) Text Transcription: h(x) = sin(x + pi/2) + 1

Given \(f(x)=\sqrt{x}\) and \(g(x)=x^{2}-1\), evaluate each expression. (a) f(g(1)) (b) g(f(1)) (c) g(f(0)) (d) f(g(-4)) (e) f(g(x)) (f) g(f(x)) Text Transcription: f(x) = sqrt x g(x) = x^2 - 1

Given f(x) = sin x and \(g(x)=\pi x\), evaluate each expression. (a) f(g(2)) (b) \(f\left(g\left(\frac{1}{2}\right)\right)\) (c) g(f(0)) (d) \(g\left(f\left(\frac{\pi}{4}\right)\right)\) (e) f(g(x)) (f) g(f(x)) Text Transcription: g(x) = pi x f(g(1/2)) g(f(pi/4))

In Exercises 61-64, find the composite functions \((f\ \circ\ g)\) and \((g\ \circ\ f)\). What is the domain of each composite function? Are the two composite functions equal? \(f(x)=x^{2}\) \(g(x)=\sqrt{x}\) Text Transcription: (f circ g) (g circ f) f(x) = x^2 g(x) = sqrt x

In Exercises 61-64, find the composite functions \((f\ \circ\ g)\) and \((g\ \circ\ f)\). What is the domain of each composite function? Are the two composite functions equal? \(f(x)=x^{2}-1\) g(x) = cos x Text Transcription: (f circ g) (g circ f) f(x) = x^2 - 1

In Exercises 61-64, find the composite functions \((f\ \circ\ g)\) and \((g\ \circ\ f)\). What is the domain of each composite function? Are the two composite functions equal? \(f(x)=\frac{3}{x}\) \(g(x)=x^{2}-1\) Text Transcription: (f circ g) (g circ f) f(x) = 3/x g(x) = x^2 - 1

In Exercises 61-64, find the composite functions \((f\ \circ\ g)\) and \((g\ \circ\ f)\). What is the domain of each composite function? Are the two composite functions equal? \(f(x)=\frac{1}{x}\) \(g(x)=\sqrt{x+2}\) Text Transcription: (f circ g) (g circ f) f(x) = 1/x g(x) = sqrt x+2

Use the graphs of f and g to evaluate each expression. If the result is undefined, explain why. (a) \((f\ \circ\ g)(3)\) (b) g(f(2)) (c) g(f(5)) (d) \((f\ \circ\ g)(-3)\) (e) \((g\ \circ\ f)(-1)\) (f) f(g(-1)) Text Transcription: (f circ g)(3) (f circ g)(-3) (g circ f)(-1)

Ripples A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius (in feet) of the outer ripple is given by r(t) = 0.6t, where t is the time in seconds after the pebble strikes the water. The area of the circle is given by the function \(A(r)=\pi r^{2}\). Find and interpret \((A\ \circ\ r)(t)\).. Text Transcription: A(r) = pi r^2 (A circ r)(t)

Think About It In Exercises 67 and 68, \(F(x)=f\ \circ\ g\ \circ\ h\). Identify functions for f, g, and h. (There are many correct answers.) \(F(x)=\sqrt{2 x-2}\) Text Transcription: F(x) = f circ g circ h F(x) = sqrt 2x-2

Think About It In Exercises 67 and 68, \(F(x)=f\ \circ\ g\ \circ\ h\). Identify functions for f, g, and h. (There are many correct answers.) F(x) = -4 sin(1 - x) Text Transcription: F(x) = f circ g circ h

In Exercises 69-72, determine whether the function is even, odd, or neither. Use a graphing utility to verify your result. \(f(x)=x^{2}\left(4-x^{2}\right)\) Text Transcription: f(x) = x^2 (4 - x^2)

In Exercises 69-72, determine whether the function is even, odd, or neither. Use a graphing utility to verify your result. \(f(x)=\sqrt[3]{x}\) Text Transcription: f(x) = cube root x

In Exercises 69-72, determine whether the function is even, odd, or neither. Use a graphing utility to verify your result. f(x) = x cos x

In Exercises 69-72, determine whether the function is even, odd, or neither. Use a graphing utility to verify your result. \(f(x)=\sin ^{2} x\) Text Transcription: f(x) = sin^2 x

Think About It In Exercises 73 and 74, find the coordinates of a second point on the graph of a function f if the given point is on the graph and the function is (a) even and (b) odd. \(\left(-\frac{3}{2},\ 4\right)\) Text Transcription: (-3/2, 4)

Think About It In Exercises 73 and 74, find the coordinates of a second point on the graph of a function f if the given point is on the graph and the function is (a) even and (b) odd. (4, 9)

The graphs of f, g, and h are shown in the figure. Decide whether each function is even, odd, or neither.

The domain of the function f shown in the figure is \(-6 \leq x \leq 6\). (a) Complete the graph of f given that f is even. (b) Complete the graph of f given that f is odd. Text Transcription: -6 leq x leq 6

Writing Functions In Exercises 77-80, write an equation for a function that has the given graph. Line segment connecting (-2, 4) and (0, -6)

Writing Functions In Exercises 77-80, write an equation for a function that has the given graph. Line segment connecting (3, 1) and (5, 8)

Writing Functions In Exercises 77-80, write an equation for a function that has the given graph. The bottom half of the parabola \(x+y^{2}=0\) Text Transcription: x + y^2 = 0

Writing Functions In Exercises 77-80, write an equation for a function that has the given graph. The bottom half of the circle \(x^{2}+y^{2}=36\) Text Transcription: x^2 + y^2 = 36

In Exercises 81-84, sketch a possible graph of the situation. The speed of an airplane as a function of time during a 5-hour flight

In Exercises 81-84, sketch a possible graph of the situation. The height of a baseball as a function of horizontal distance during a home run

In Exercises 81-84, sketch a possible graph of the situation. The amount of a certain brand of sneaker sold by a sporting goods store as a function of the price of the sneaker

In Exercises 81-84, sketch a possible graph of the situation. The value of a new car as a function of time over a period of 8 years

Find the value of c such that the domain of \(f(x)=\sqrt{c-x^{2}}\) is [-5, 5]. Text Transcription: f(x) = sqrt c-x^2

Find all values of c such that the domain of \(f(x)=\frac{x+3}{x^{2}+3 c x+6}\) is the set of all real numbers. Text Transcription: f(x) = x+3 / x^2 +3cx+6

Graphical Reasoning An electronically controlled thermostat is programmed to lower the temperature during the night automatically (see figure). The temperature T in degrees Celsius is given in terms of t, the time in hours on a 24-hour clock. (a) Approximate T(4) and T(15). (b) The thermostat is reprogrammed to produce a temperature H(t) = T(t - 1). How does this change the temperature? Explain. (c) The thermostat is reprogrammed to produce a temperature H(t) = T(t) - 1. How does this change the temperature? Explain.

Water runs into a vase of height 30 centimeters at a constant rate. The vase is full after 5 seconds. Use this information and the shape of the vase shown to answer the questions if d is the depth of the water in centimeters and t is the time in seconds (see figure). (a) Explain why d is a function of t. (b) Determine the domain and range of the function. (c) Sketch a possible graph of the function. (d) Use the graph in part (c) to approximate d(4). What does this represent?

Modeling Data The table shows the average numbers of acres per farm in the United States for selected years. (Source: U.S. Department of Agriculture) (a) Plot the data, where A is the acreage and t is the time in years, with t = 5 corresponding to 1955. Sketch a freehand curve that approximates the data. (b) Use the curve in part (a) to approximate A(20).

Automobile Aerodynamics The horsepower H required to overcome wind drag on a certain automobile is approximated by \(H(x)=0.002x^2+0.005x-0.029,\quad\ \ 10\le x\le100\) where x is the speed of the car in miles per hour. (a) Use a graphing utility to graph H. (b) Rewrite the power function so that x represents the speed in kilometers per hour. [Find H(x/1.6).] Text Transcription: H(x) = 0.002x^2 + 0.005x - 0.029, 10 leq x leq 100

Think About It Write the function f(x) = |x| + |x – 2| without using absolute value signs. (For a review of absolute value, see Appendix C.)

Writing Use a graphing utility to graph the polynomial functions \(p_{1}(x)=x^{3}-x+1\) and \(p_{2}(x)=x^{3}-x\). How many zeros does each function have? Is there a cubic polynomial that has no zeros? Explain. Text Transcription: p_1 (x) = x^3 - x + 1 p_2 (x) = x^3 - x

Prove that the function is odd. \(f(x)=a_{2 n+1} x^{2 n+1}+\cdots+a_{3} x^{3}+a_{1} x\) Text Transcription: f(x) = a_2n+1 x^2n+1 + cdot + a_3 x^3 + a_1 x

Prove that the function is even. \(f(x)=a_{2 n} x^{2 n}+a_{2 n-2} x^{2 n-2}+\cdots+a_{2} x^{2}+a_{0}\) Text Transcription: f(x) = a_2n x^2n + a_2n-2 x^2n-2 + cdot + a_2 x^2 + a_0

Prove that the product of two even (or two odd) functions is even.

Prove that the product of an odd function and an even function is odd.

Volume An open box of maximum volume is to be made from a square piece of material 24 centimeters on a side by cutting equal squares from the corners and turning up the sides (see figure). (a) Write the volume V as a function of x, the length of the corner squares. What is the domain of the function? (b) Use a graphing utility to graph the volume function and approximate the dimensions of the box that yield a maximum volume. (c) Use the table feature of a graphing utility to verify your answer in part (b). (The first two rows of the table are shown.)

Length A right triangle is formed in the first quadrant by the x- and y-axes and a line through the point (3, 2) (see figure). Write the length L of the hypotenuse as a function of x.

True or False? In Exercises 99-102, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If f(a) = f(b), then a = b.

True or False? In Exercises 99-102, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. A vertical line can intersect the graph of a function at most once.

True or False? In Exercises 99-102, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If f(x) = f(-x) for all x in the domain of f, then the graph of f is symmetric with respect to the y-axis.

True or False? In Exercises 99-102, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If f is a function, then f(ax) = af(x).

Let R be the region consisting of the points (x, y) of the Cartesian plane satisfying both \(|x|-|y| \leq 1\) and \(|y| \leq 1\). Sketch the region R and find its area. Text Transcription: |x| - |y| leq 1 |y| leq 1

Consider a polynomial f(x) with real coefficients having the property f(g(x)) = g(f(x)) for every polynomial g(x) with real coefficients. Determine and prove the nature of f(x).