Calculus: Early Transcendentals 1st Edition solutions

Calculus: Early Transcendentals 1
Explain how to find the domain of \(f\ \circ\ g\) if you know the domain and range of f and g.

Calculus: Early Transcendentals 1
19-20. Domain in context Determine an appropriate domain of each function. Identify the independent and dependent variables. A stone is thrown vertically upward from the ground at a speed of 40 m/s at time t = 0. Its distance d (in m) above the ground (neglecting air resistance) is approximated b

Calculus: Early Transcendentals 1
19-20. Domain in context Determine an appropriate domain of each function. Identify the independent and dependent variables. The average production cost for a company to make n bicycles is given by the function c(n) = 120 - 0.25n.

Calculus: Early Transcendentals 1
Composite functions from graphs Use the graphs of f and g in the figure to determine the following function values. a. f(g(2)) b. g(f(2)) c. f(g(4)) d. g(f(5)) e. f(g(7)) f. f(f(8))

Calculus: Early Transcendentals 1
Composite functions from tables Use the table to evaluate the given compositions. a. h(g(0)) b. g(f(4)) c. h(h(0)) d. g(h(f(4))) e. f(f(f(1))) f. h(h(h(0))) g. f(h(g(2))) h. g(f(h(4)

Calculus: Early Transcendentals 1
Absolute value graphs Use the definition of absolute value to graph the equation |x| - |y| = 1. Use a graphing utility only to check your work.

Calculus: Early Transcendentals 1
Use the terms domain, range, independent variable, and dependent variable to explain how a function relates one variable to another variable.

Calculus: Early Transcendentals 1
Does the independent variable of a function belong to the domain or range? Does the dependent variable belong to the domain or range?

Calculus: Early Transcendentals 1
Explain how the vertical line test is used to detect functions.

Calculus: Early Transcendentals 1
Find possible choices for outer and inner functions and such that the given functions equals .Give the domain of .

Calculus: Early Transcendentals 1
If \(f(x)=1 /\left(x^{3}+1\right)\), what is f(2)? What is \(f\left(y^{2}\right)\)?

Calculus: Early Transcendentals 1
Which statement about a function is true? (i) For each value of x in the domain, there corresponds one value of y; (ii) for each value of y in the range, there corresponds one value of x. Explain.

Calculus: Early Transcendentals 1
If \(f(x)=\sqrt{x}\) and \(g(x)=x^{3}-2\), find the compositions \(f\ \circ\ g,\ g\ \circ\ f,\ f\ \circ\ f\) and \(g\ \circ\ g\)

Calculus: Early Transcendentals 1
If \(f(\pm 2)=2\) and \(g(\pm 2)=-2\), evaluate f(g(2)) and g(f(-2))

Calculus: Early Transcendentals 1
Sketch a graph of an even function and give the function's defining property.

Calculus: Early Transcendentals 1
Sketch a graph of an odd function and give the function's defining property

Calculus: Early Transcendentals 1
11-12. Vertical line test Decide whether graph A, graph B, or both graphs represent functions.

Calculus: Early Transcendentals 1
11-12. Vertical line test Decide whether graph A, graph B, or both graphs represent functions.

Calculus: Early Transcendentals 1
13-18. Domain and range Graph each function with a graphing utility using the given window. Then state the domain and range of the function. \(f(x)=3x^4-10;\quad\ \ [-2,\ 2]\times[-10,\ 15]\)

Calculus: Early Transcendentals 1
13-18. Domain and range Graph each function with a graphing utility using the given window. Then state the domain and range of the function. \(g(y)=\frac{y+1}{y^2-y-6};\quad\ \ [-4,\ 6]\times[-3,\ 3]\)

Calculus: Early Transcendentals 1
13-18. Domain and range Graph each function with a graphing utility using the given window. Then state the domain and range of the function. \(f(x)=\sqrt{4-x^2};\quad\ \ [-4,\ 4]\times[-4,\ 4]\)

Calculus: Early Transcendentals 1
13-18. Domain and range Graph each function with a graphing utility using the given window. Then state the domain and range of the function. \(F(w)=\sqrt[4]{2-w};\quad\ \ [-3,\ 2]\times[0,\ 2]\)

Calculus: Early Transcendentals 1
13-18. Domain and range Graph each function with a graphing utility using the given window. Then state the domain and range of the function. \(h(u)=\sqrt[3]{u-1};\ \ \quad[-7,\ 9]\times[-2,\ 2]\)

Calculus: Early Transcendentals 1
13-18. Domain and range Graph each function with a graphing utility using the given window. Then state the domain and range of the function. \(g(x)=\left(x^2-4\right)\sqrt{x+5};\quad\ \ [-5,\ 5]\times[-10,\ 50]\)

Calculus: Early Transcendentals 1
21-30. Composite functions and notation Let \(f(x)=x^2-4,\ g(x)=x^3,\) and F(x) = 1/(x - 3). Simplify or evaluate the following expressions. f(10)

Calculus: Early Transcendentals 1
21-30. Composite functions and notation Let \(f(x)=x^2-4,\ g(x)=x^3,\) and F(x) = 1/(x - 3). Simplify or evaluate the following expressions. \(f\left(p^{2}\right)\)

Calculus: Early Transcendentals 1
21-30. Composite functions and notation Let \(f(x)=x^2-4,\ g(x)=x^3,\) and F(x) = 1/(x - 3). Simplify or evaluate the following expressions. g(1/z)

Calculus: Early Transcendentals 1
21-30. Composite functions and notation Let \(f(x)=x^2-4,\ g(x)=x^3,\) and F(x) = 1/(x - 3). Simplify or evaluate the following expressions. \(F\left(y^{4}\right)\)

Calculus: Early Transcendentals 1
21-30. Composite functions and notation Let \(f(x)=x^2-4,\ g(x)=x^3,\) and F(x) = 1/(x - 3). Simplify or evaluate the following expressions. F(g(y))

Calculus: Early Transcendentals 1
21-30. Composite functions and notation Let \(f(x)=x^2-4,\ g(x)=x^3,\) and F(x) = 1/(x - 3). Simplify or evaluate the following expressions. F(g(w))

Calculus: Early Transcendentals 1
21-30. Composite functions and notation Let \(f(x)=x^2-4,\ g(x)=x^3,\) and F(x) = 1/(x - 3). Simplify or evaluate the following expressions. g(f(u))

Calculus: Early Transcendentals 1
21-30. Composite functions and notation Let \(f(x)=x^2-4,\ g(x)=x^3,\) and F(x) = 1/(x - 3). Simplify or evaluate the following expressions. \(\frac{f(2+h)-f(2)}{h}\)

Calculus: Early Transcendentals 1
21-30. Composite functions and notation Let \(f(x)=x^2-4,\ g(x)=x^3,\) and F(x) = 1/(x - 3). Simplify or evaluate the following expressions. F(F(x))

Calculus: Early Transcendentals 1
21-30. Composite functions and notation Let \(f(x)=x^2-4,\ g(x)=x^3,\) and F(x) = 1/(x - 3). Simplify or evaluate the following expressions. g(F(f(x)))

Calculus: Early Transcendentals 1
31-34. Working with composite functions Find possible choices for outer and inner functions f and g such that the given function h equals \(f\ \circ\ g\). Give the domain of h. \(h(x)=\left(x^{3}-5\right)^{10}\)

Calculus: Early Transcendentals 1
31-34. Working with composite functions Find possible choices for outer and inner functions f and g such that the given function h equals \(f\ \circ\ g\). Give the domain of h. \(h(x)=2 /\left(x^{6}+x^{2}+1\right)^{2}\)

Calculus: Early Transcendentals 1
31-34. Working with composite functions Find possible choices for outer and inner functions f and g such that the given function h equals \(f\ \circ\ g\). Give the domain of h. \(h(x)=\frac{1}{\sqrt{x^{3}-1}}\)

Calculus: Early Transcendentals 1
35-40. More composite functions Let f(x) = |x|, \(g(x)=x^{2}-4\), \(F(x)=\sqrt{x}\), and G(x) = 1/(x - 2). Determine the following composite functions and give their domains. \(f\ \circ\ g\)

Calculus: Early Transcendentals 1
35-40. More composite functions Let f(x) = |x|, \(g(x)=x^{2}-4\), \(F(x)=\sqrt{x}\), and G(x) = 1/(x - 2). Determine the following composite functions and give their domains. \(g\ \circ\ f\)

Calculus: Early Transcendentals 1
35-40. More composite functions Let f(x) = |x|, \(g(x)=x^{2}-4\), \(F(x)=\sqrt{x}\), and G(x) = 1/(x - 2). Determine the following composite functions and give their domains. \(f\ \circ\ G\)

Calculus: Early Transcendentals 1
35-40. More composite functions Let f(x) = |x|, \(g(x)=x^{2}-4\), \(F(x)=\sqrt{x}\), and G(x) = 1/(x - 2). Determine the following composite functions and give their domains. \(f\ \circ\ g\ \circ\ G\)

Calculus: Early Transcendentals 1
35-40. More composite functions Let f(x) = |x|, \(g(x)=x^{2}-4\), \(F(x)=\sqrt{x}\), and G(x) = 1/(x - 2). Determine the following composite functions and give their domains. \(G\ \circ\ g\ \circ\ f\)

Calculus: Early Transcendentals 1
35-40. More composite functions Let f(x) = |x|, \(g(x)=x^{2}-4\), \(F(x)=\sqrt{x}\), and G(x) = 1/(x - 2). Determine the following composite functions and give their domains. \(F\ \circ\ g\ \circ\ g\)

Calculus: Early Transcendentals 1
41-44. Missing piece Let \(\text { Let } g(x)=x^{2}+3\) and find a function f that produces the given composition. \((f\ \circ\ g)(x)=x^4+6x^2+9\)

Calculus: Early Transcendentals 1
41-44. Missing piece Let \(\text { Let } g(x)=x^{2}+3\) and find a function f that produces the given composition. \((f\ \circ\ g)(x)=x^4+6x^2+20\)

Calculus: Early Transcendentals 1
41-44. Missing piece Let \(\text { Let } g(x)=x^{2}+3\) and find a function f that produces the given composition. \((g\ \circ\ f)(x)=x^4+3\)

Calculus: Early Transcendentals 1
41-44. Missing piece Let \(\text { Let } g(x)=x^{2}+3\) and find a function f that produces the given composition. \((g\ \circ\ f)(x)=x^{2/3}+3\)

Calculus: Early Transcendentals 1
47-52. Symmetry Determine whether the graphs of the following equations and functions have symmetry about the x-axis, the y-axis, or the origin. Check your work by graphing. \(f(x)=x^{4}+5 x^{2}-12\)

Calculus: Early Transcendentals 1
47-52. Symmetry Determine whether the graphs of the following equations and functions have symmetry about the x-axis, the y-axis, or the origin. Check your work by graphing. \(f(x)=3 x^{5}+2 x^{3}-x\)

Calculus: Early Transcendentals 1
47-52. Symmetry Determine whether the graphs of the following equations and functions have symmetry about the x-axis, the y-axis, or the origin. Check your work by graphing. \(f(x)=x^{5}-x^{3}-2\)

Calculus: Early Transcendentals 1
47-52. Symmetry Determine whether the graphs of the following equations and functions have symmetry about the x-axis, the y-axis, or the origin. Check your work by graphing. f(x) = 2|x|

Calculus: Early Transcendentals 1
47-52. Symmetry Determine whether the graphs of the following equations and functions have symmetry about the x-axis, the y-axis, or the origin. Check your work by graphing. \(x^{2 / 3}+y^{2 / 3}=1\)

Calculus: Early Transcendentals 1
47-52. Symmetry Determine whether the graphs of the following equations and functions have symmetry about the x-axis, the y-axis, or the origin. Check your work by graphing. \(x^{3}-y^{5}=0\)

Calculus: Early Transcendentals 1
Symmetry in graphs State whether the functions represented by graphs A, B, and C in the figure are even, odd, or neither.

Calculus: Early Transcendentals 1
Symmetry in graphs State whether the functions represented by graphs A, B, and C in the figure are even, odd, or neither.

Calculus: Early Transcendentals 1
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The range of f(x) = 2x - 38 is all real numbers. b. The relation \(f(x)=x^{6}+1\) is not a function because f(1) = f(-1) = 2 c. If \(f(x)=x^{-1}\), then f(1/x) = 1/f(x)

Calculus: Early Transcendentals 1
Range of power functions Using words and figures, explain why the range of \(f(x)=x^{n}\), where n is a positive odd integer, is all real numbers. Explain why the range of \(g(x)=x^{n}\), where n is a positive even integer, is all nonnegative real numbers.

Calculus: Early Transcendentals 1
Even and odd at the origin a. If f(0) is defined and f is an even function, is it necessarily true that f(0) = 0? Explain. b. If f(0) is defined and f is an odd function, is it necessarily true that f(0) = 0? Explain.

Calculus: Early Transcendentals 1
59-62. Polynomial composition Determine a polynomial f that satisfies the following properties. (Hint: Determine the degree of f; then substitute a polynomial of that degree and solve for its coefficients.) f(f(x)) = 9x - 8

Calculus: Early Transcendentals 1
59-62. Polynomial composition Determine a polynomial f that satisfies the following properties. (Hint: Determine the degree of f; then substitute a polynomial of that degree and solve for its coefficients.) \((f(x))^{2}=9 x^{2}-12 x+4\)

Calculus: Early Transcendentals 1
59-62. Polynomial composition Determine a polynomial f that satisfies the following properties. (Hint: Determine the degree of f; then substitute a polynomial of that degree and solve for its coefficients.) \(f(f(x))=x^{4}-12 x^{2}+30\)

Calculus: Early Transcendentals 1
59-62. Polynomial composition Determine a polynomial f that satisfies the following properties. (Hint: Determine the degree of f; then substitute a polynomial of that degree and solve for its coefficients.) \((f(x))^{2}=x^{4}-12 x^{2}+36\)

Calculus: Early Transcendentals 1
Launching a rocket A small rocket is launched vertically upward from the edge of a cliff 80 ft off the ground at a speed of 96 ft/s. Its height above the ground is given by the function \(h(t)=-16 t^{2}+96 t+80\), where t represents time measured in seconds. a. Assuming the rocket is launched at

Calculus: Early Transcendentals 1
Draining a tank (Torricelli's law) A cylindrical tank with a cross-sectional area of \(100\mathrm{\ cm}^2\) is filled to a depth of 100 cm with water. At t = 0, a drain in the bottom of the tank with an area of \(10\mathrm{\ cm}^2\) is opened allowing water to flow out of the tank. The depth of water

Calculus: Early Transcendentals 1
65-71. Combining even and odd functions Let E be an even function and O be an odd function. Determine the symmetry, if any, of the following functions. E + O

Calculus: Early Transcendentals 1
65-71. Combining even and odd functions Let E be an even function and O be an odd function. Determine the symmetry, if any, of the following functions. \(E \cdot O\)

Calculus: Early Transcendentals 1
65-71. Combining even and odd functions Let E be an even function and O be an odd function. Determine the symmetry, if any, of the following functions. E/O

Calculus: Early Transcendentals 1
65-71. Combining even and odd functions Let E be an even function and O be an odd function. Determine the symmetry, if any, of the following functions. \(E \ \circ \ O\)

Calculus: Early Transcendentals 1
65-71. Combining even and odd functions Let E be an even function and O be an odd function. Determine the symmetry, if any, of the following functions. \(E \ \circ \ E\)

Calculus: Early Transcendentals 1
65-71. Combining even and odd functions Let E be an even function and O be an odd function. Determine the symmetry, if any, of the following functions. \(O \ \circ \ O\)

Calculus: Early Transcendentals 1
65-71. Combining even and odd functions Let E be an even function and O be an odd function. Determine the symmetry, if any, of the following functions. \(O \ \circ \ E\)

Calculus: Early Transcendentals 1
72-75. Working with function notation Consider the following functions and simplify the expressions \(\frac{f(x)-f(a)}{x-a}\) and \(\frac{f(x+h)-f(x)}{h}\). f(x) = 3 - 2x

Calculus: Early Transcendentals 1
72-75. Working with function notation Consider the following functions and simplify the expressions \(\frac{f(x)-f(a)}{x-a}\) and \(\frac{f(x+h)-f(x)}{h}\). f(x) = 4x - 3

Calculus: Early Transcendentals 1
72-75. Working with function notation Consider the following functions and simplify the expressions \(\frac{f(x)-f(a)}{x-a}\) and \(\frac{f(x+h)-f(x)}{h}\). \(f(x)=4 x^{2}-1\)

Calculus: Early Transcendentals 1
72-75. Working with function notation Consider the following functions and simplify the expressions \(\frac{f(x)-f(a)}{x-a}\) and \(\frac{f(x+h)-f(x)}{h}\). f(x) = 1/(2x)