Question: One of the fundamental themes of calculus

> Test for symmetry with respect to each axis and to the origin.

> Can the graph of a one-to-one function intersect a horizontal line more than once? Explain.

> Find all values of c such that the domain of f(x) = x + 3 / x2 + 3cx + 6

> Find the value of c such that the domain of f(x) = √c − x2 is [−5, 5].

> A person buys a new car and keeps it for 6 years. During year 4, he buys several expensive upgrades. Consider the value of the car as a function of time.

> A student commutes 15 miles to attend college. After driving for a few minutes, she remembers that a term paper that is due has been forgotten. Driving faster than usual, she returns home, picks up the paper, and once again starts toward school. Consider

> The height of a baseball as a function of horizontal distance during a home run

> Sketch a possible graph of the situation. The speed of an airplane as a function of time during a 5-hour flight

> A large room contains two speakers that are 3 meters apart. The sound intensity I of one speaker is twice that of the other, as shown in the figure. (To print an enlarged copy of the graph, go to MathGraphs.com.) Suppose the listener is free to move abou

> Write an equation for a function that has the given graph. The bottom half of the circle x2 + y2 = 36

> Write an equation for a function that has the given graph. The bottom half of the parabola x + y2 = 0

> Test for symmetry with respect to each axis and to the origin. xy − √4 – x2 = 0

> Write an equation for a function that has the given graph. Line segment connecting (3, 1) and (5, 8)

> Write an equation for a function that has the given graph. Line segment connecting (−2, 4) and (0, −6)

> Determine whether the function is even, odd, or neither. Then find the zeros of the function. Use a graphing utility to verify your result. f(x) = 4x4 − 3x2

> Determine whether the function is even, odd, or neither. Then find the zeros of the function. Use a graphing utility to verify your result. f(x) = 2 6√x

> Determine whether the function is even, odd, or neither. Then find the zeros of the function. Use a graphing utility to verify your result. f(x) = ∛x

> Determine whether the function is even, odd, or neither. Then find the zeros of the function. Use a graphing utility to verify your result. f(x) = x2 (4 − x2 )

> Explain how you would graph the equation y + ∣y∣ = x + ∣x∣. Then sketch the graph.

> The domain of the function f shown in the figure is −6 ≤ x ≤ 6. a. Complete the graph of f given that f is even. b. Complete the graph of f given that f is odd.

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

> Find the coordinates of a second point on the graph of a function f when the given point is on the graph and the function is (a) even and (b) odd. (4, 9)

> Test for symmetry with respect to each axis and to the origin. y = 4 − √x+ 3

> Find the coordinates of a second point on the graph of a function f when the given point is on the graph and the function is (a) even and (b) odd. (-3/2, 4 )

> F(x) = f ∘ g ∘ h. Identify functions for f, g, and h. (There are many correct answers.) F(x) = 1 / 4x6

> F(x) = f ∘ g ∘ h. Identify functions for f, g, and h. (There are many correct answers.) F(x) = √2x − 2

> 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

> Use the graphs of f and g to evaluate each expression. If the result is undefined, explain why. a. ( f ∘ g)(3) b. g(f(2)) c. g(f(5)) d. (f ∘ g)(−3) e. (g &

> Let f(x) = 1/ 1-x. a. What are the domain and range of f? b. Find the composition f( f(x)). What is the domain of this function? c. Find f( f( f(x))). What is the domain of this function? d. Graph f( f( f(x))). Is the graph a line? Why or why not?

> Find the composite functions f ∘ g and g ∘ f. Find the domain of each composite function. Are the two composite functions equal? f(x) = 1/x g(x) = √x + 2

> Find the composite functions f ∘ g and g ∘ f. Find the domain of each composite function. Are the two composite functions equal? f(x) = 3/x g(x) = x2 − 1

> Find the composite functions f ∘ g and g ∘ f. Find the domain of each composite function. Are the two composite functions equal? f(x) = x2 – 1 g(x) = −x

> Find the composite functions f ∘ g and g ∘ f. Find the domain of each composite function. Are the two composite functions equal? f(x) = x2 g(x) = √x

> Test for symmetry with respect to each axis and to the origin. xy2 = −10

> Given f(x) = 2x3 and g(x) = 4x + 3, evaluate each expression. a. f(g(0)) b. f (g( 1 2)) c. g(f(0)) d. g( f (−1 4)) e. f(g(x)) f. g( f(x))

> Given f(x) = √x and g(x) = x2 − 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))

> Find (a) f (x) + g(x), (b) f (x) − g(x), (c) f (x) ● g(x), and (d) f (x) g(x). f(x) = x2 + 5x + 4 g(x) = x + 1

> Find (a) f (x) + g(x), (b) f (x) − g(x), (c) f (x) ∙ g(x), and (d) f (x) g(x). f(x) = 2x – 5 g(x) = 4 − 3x

> Sketch the graph of the function f(x) = √x and label the point (4, 2) on the graph. a. Find the slope of the line joining (4, 2) and (9, 3). Is the slope of the tangent line at (4, 2) greater than or less than this number? b. Find the slope of the line j

> 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 MathGraphs.com. a. f(x − 4) b. f(x + 2) c. f(x) + 4 d. f(x) − 1 e. 2f(x) f. 1 2 f(x) g. f

> 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 MathGraphs.com. a. f(x + 3) b. f(x − 1) c. f(x) + 2 d. f(x) − 4 e. 3f(x) f. 1/4 f(x) g

> Use the graph of y = f (x) to match the function with its graph. y = f(x − 1) + 3

> Use the graph of y = f (x) to match the function with its graph. y = f(x + 6) + 2

> Use the graph of y = f (x) to match the function with its graph. y = −f(x − 4)

> Test for symmetry with respect to each axis and to the origin. xy = 4

> Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the domain of a function consists of a single number, then its range must also consist of only one number.

> Use the graph of y = f (x) to match the function with its graph. y = −f(−x) − 2

> Use the graph of y = f (x) to match the function with its graph. y = f(x) − 5

> Use the graph of y = f (x) to match the function with its graph. y = f(x + 5)

> The graph shows one of the six basic functions and a transformation of the function. Describe the transformation. Then use your description to write an equation for the transformation.

> The graph shows one of the six basic functions and a transformation of the function. Describe the transformation. Then use your description to write an equation for the transformation.

> The graph shows one of the six basic functions and a transformation of the function. Describe the transformation. Then use your description to write an equation for the transformation.

> The graph shows one of the six basic functions and a transformation of the function. Describe the transformation. Then use your description to write an equation for the transformation.

> Determine whether y is a function of x. x2 y − x2 + 4y = 0

> Determine whether y is a function of x. y2 = x2 − 1

> Test for symmetry with respect to each axis and to the origin. y = x3 + x

> Sketch the graph of the function. y = -4 csc 3x

> Sketch the graph of the function. y = -sec 2πx

> Sketch a graph of the function and find its domain and range. Use a graphing utility to verify your graph. h(x) = √x − 6

> Sketch the graph of the function. y = cot x/2

> Sketch the graph of the function. y = 1/3 tan x

> Sketch the graph of the function. y = 8 cos x/4

> Sketch the graph of the function. y = 3 sin 2x/5

> Sketch the graph of the function. y = sin πx

> Determine whether y is a function of x. x2 + y = 16

> Sketch the graph of the function. y = 9 cos x

> Solve the equation for θ, where 0 ≤ θ ≤ 2π 2 sec2 θ + tan2 θ − 5 = 0

> Evaluate the function at the given value(s) of the independent variable. Simplify the results. f(x) = 3x – 1 f(x) – f(1) / x - 1

> Solve the equation for θ, where 0 ≤ θ ≤ 2π sec2 θ − sec θ − 2 = 0

> You drive to the beach at a rate of 120 kilometers per hour. On the return trip, you drive at a rate of 60 kilometers per hour. What is your average speed for the entire trip? Explain your reasoning.

> Solve the equation for θ, where 0 ≤ θ ≤ 2π cos3 θ = cos θ

> Solve the equation for θ, where 0 ≤ θ ≤ 2π 2 sin2 θ + 3 sin θ + 1 = 0

> solve the equation for θ, where 0 ≤ θ ≤ 2π 2 cos2 θ = 1

> solve the equation for θ, where 0 ≤ θ ≤ 2π 2 cos θ + 1 = 0

> Use a calculator to evaluate the trigonometric function. Round your answers to four decimal places. cos (–3π / 7)

> Use a calculator to evaluate the trigonometric function. Round your answers to four decimal places. sin (–π / 9)

> Use a calculator to evaluate the trigonometric function. Round your answers to four decimal places. csc 2π / 9

> Determine whether y is a function of x. x2 + y2 = 16

> Use a calculator to evaluate the trigonometric function. Round your answers to four decimal places. sec 12π / 5

> Test for symmetry with respect to each axis and to the origin. y2 = x3 − 8x

> You are in a boat 2 miles from the nearest point on the coast. You will travel to a point Q located 3 miles down the coast and 1 mile inland (see figure). You can row at 2 miles per hour and walk at 4 miles per hour. Write the total time T of the trip as

> Use a calculator to evaluate the trigonometric function. Round your answers to four decimal places. cot 401º

> Use a calculator to evaluate the trigonometric function. Round your answers to four decimal places. tan 33º

> Evaluate the sine, cosine, and tangent of the angle. Do not use a calculator. tan 33º

> Evaluate the sine, cosine, and tangent of the angle. Do not use a calculator. 405º

> Evaluate the sine, cosine, and tangent of the angle. Do not use a calculator. -4π/3

> Evaluate the sine, cosine, and tangent of the angle. Do not use a calculator. 13π/6

> Evaluate the sine, cosine, and tangent of the angle. Do not use a calculator. 240º

> Evaluate the sine, cosine, and tangent of the angle. Do not use a calculator. -45º

> Convert the radian measure to degree measure. -13π / 6

> Vertical Line Test to determine whether y is a function of x. To print an enlarged copy of the graph, go to MathGraphs.com. x2 + y2 = 4

> Find any intercepts. y = 4x2 + 3

> Sketch a graph of the function and find its domain and range. Use a graphing utility to verify your graph. f(t) = 2 / 7 + t

> Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

> Use a graphing utility to compare the graph of f(x) = 4/ π (sin πx + 1/3 sin 3 πx) with the given graph. Try to improve the approximation by adding a term to f(x). Use a graphing utility to verify that your new approximat

> The monthly sales S (in thousands of units) of a seasonal product are modeled by S = 58.3 + 32.5 cos πt/6 where t is the time (in months), with t = 1 corresponding to January. Use a graphing utility to graph the model for S and determine the months when

> The model for the height h of a Ferris wheel car is h = 51 + 50 sin 8πt where t is measured in minutes. (The Ferris wheel has a radius of 50 feet.) This model yields a height of 51 feet when t = 0. Alter the model so that the height of the car is 1 foot

> Sketch the graphs of f(x) = sin x, g(x) = │sin x│, and h(x) = sin(│x│). In general, how are the graphs of ∣f(x)∣ and f(∣x∣) related to the graph of f ?

> Consider an angle in standard position with r = 12 centimeters, as shown in the figure. Describe the changes in the values of x, y, sin θ, cos θ, and tan θ as θ increases continually from 0°

> How do the ranges of the cosine function and the secant function compare?