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study help
mathematics
precalculus
Questions and Answers of
Precalculus
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If y = f(x)g(x), then dy/dx = f′(x)g′(x).
Graph the function. f(x) = Jx, 12 - (2 - x, 0≤x≤ 1 1 < x < 2
Tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling
Evaluate cos π/12.
Copy and complete the following table of function values. If the function is undefined at a given angle, enter “UND.” Do not use a calculator or tables. 0 sin 0 cos tan 0 cot 0 sec 0 csc -T -
Which of the graphs are graphs of functions of x, and which are not? Give reasons for your answers.a.b. X- 0
One of sin x, cos x, and tan x is given. Find the other two if x lies in the specified interval. sin x -3. 5' XE TT 2. ㅠ
On a circle of radius 10 m, how long is an arc that subtends a central angle of (a) 4π/5 radians? (b) 110°?
Write a formula for ƒ ∘ g ∘ h. f(x)=√x + 1, g(x) = 1 x + 4' h(x) = 1 X
One of sin x, cos x, and tan x is given. Find the other two if x lies in the specified interval. tan x -1/10 2² = xe]TT, 3πT 2
Find the domains and ranges of ƒ, g, ƒ/g, and g/ƒ.ƒ(x) = 2, g(x) = x2 + 1
You want to make an 80° angle by marking an arc on the perimeter of a 12-in.-diameter disk and drawing lines from the ends of the arc to the disk’s center. To the nearest tenth of an inch, how
One of sin x, cos x, and tan x is given. Find the other two if x lies in the specified interval. COS X = 3' -|- NE 2
Evaluate each expression using the given table of values:a. ƒ(g(-1)) b. g(ƒ(0)) c. ƒ(ƒ(-1))d. g(g(2)) e. g(ƒ(-2)) f. ƒ(g(1)) X f(x) g(x) -2 −1 1 2 0 1 0 -2 0 1 1 - 1 2 2 0
Copy and complete the following table. g(x) a. X 7 b. x + 2 c. ? d. X x - 1 e. ? 1 X f. f(x) Vx 3x √x - 5 X X - 1 + ? 1 X (fog)(x) ? ? √x²-5 ? X X
Express the area and perimeter of an equilateral triangle as a function of the triangle’s side length x.
What is the period of each function? -sin TTX 3
Let ƒ(x) = x - 3, g(x) = √x, h(x) = x3, and j(x) = 2x. Express each of the functions as a composite involving one or more of ƒ, g, h, and j.a. y = √x - 3 b. y = 2√xc. y = x1/4 d. y
(a) Write formulas for ƒ ∘ g and g ∘ ƒ and find the (b) Domain and (c) Range of each. f(x) = √x + 1, g(x) 1 X
Express the edge length of a cube as a function of the cube’s diagonal length d. Then express the surface area and volume of the cube as a function of the diagonal length.
A point P in the first quadrant lies on the graph of the function ƒ(x) = √x. Express the coordinates of P as functions of the slope of the line joining P to the origin.
Find the natural domain and graph the function. g(x) = √|x|
What is the period of each function? sin x 1 TT 4 + 1
What is the period of each function?sin 2x
The accompanying figure shows the graph of y = -x2 shifted to two new positions. Write equations for the new graphs. -7 Position (a) y = -x² 0 4 Position (b) X
What is the period of each function?cos πx
What is the period of each function? COS X TT 2
LetFind a function y = g(x) so that (ƒ ∘ g)(x) = x. f(x) = X x 21
Find the natural domain and graph the function.ƒ(x) = 5 - 2x
Graph the function. The ts-plane (t-axis horizontal, s-axis vertical). What is the period of each function? What symmetries do the graphs have? S = sec πt 2
Match the equation listed in parts (a)–(d) to the graphs in the accompanying figure.a. y = (x - 1)2 - 4 b. y = (x - 2)2 + 2c. y = (x + 2)2 + 2 d. y = (x + 3)2 - 2 Position
Find the natural domain and graph the function.F(t) = t/|t|
Graph the function F(x) = (4- x², x² + 2x, x ≤ 1 x > 1
Find a formula for each function graphed.a.b. 1 y 0 (1, 1) 2 → X
Graph the function. The ts-plane (t-axis horizontal, s-axis vertical). What is the period of each function? What symmetries do the graphs have?s = cot 2t
Graph the following equation and explain why they are not graphs of functions of x.a. |y| = x b. y2 = x2
Tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling
Find a formula for each function graphed.a.b. У (-1, 1) (1, 1) 10 3 X
Graph y = sin x and y = [sin x] together. What are the domain and range of [sin x]?
Tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling
Express the given quantity in terms of sin x and cos x. sin З п 2 X
Tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling
Evaluate cos π/12. 7TT sin as sin 12 6|4 + TT TT 3
Tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling
Graph the functions.y = |x - 2|
What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.
What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing. y = √ [x]
Express the given quantity in terms of sin x and cos x.cos (π + x)
Graph the function.y = 1 + √x - 1
What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.
Graph the function. y 1 x - 2
Graph the function.y = (x + 1)2/3
Graph the function. y = = + 2 X
Graph the function. y 1 (x - 1)²
Graph the function.y = 1 - x2/3
What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.y = x3/8
Graph the function.y = 23x - 1 - 1
What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.y = -x3/2
Find the function value.cos2 π/8
The accompanying figure shows the graph of a function ƒ(x) with domain [0, 2] and range [0, 1]. Find the domains and ranges of the following functions, and sketch their graphs.a. ƒ(x) + 2 b.
Find the function value.sin2 π/12
Foridentify A, B, C, and D for the sine functions and sketch their graphs. f(x) = A sin ( ² 7 7 (x − c ) ) + - C) + D.
Solve for the angle θ, where 0 ≤ θ ≤ 2π.sin2 θ = 3/4
Solve for the angle θ, where 0 ≤ θ ≤ 2π.sin 2θ - cos θ = 0
Foridentify A, B, C, and D for the sine functions and sketch their graphs.y = 2sin(x + π) - 1 f(x) = A sin ( ² 7 7 (x − c ) ) + - C) + D.
Tell by what factor and direction the graphs of the given functions are to be stretched or compressed. Give an equation for the stretched or compressed graph.y = 1 + 1/x2, compressed vertically by a
A triangle has sides a = 2 and b = 3 and angle C = 60°. Find the length of side c.
Tell by what factor and direction the graphs of the given functions are to be stretched or compressed. Give an equation for the stretched or compressed graph.y = √4 - x2, stretched horizontally by
A triangle has side c = 2 and angles A = π/4 and B = π/3. Find the length a of the side opposite A.
Graph the function y = |x2 - 1|.
Assume that ƒ is an even function, g is an odd function, and both ƒ and g are defined on the entire real line (-∞, ∞). Which of the following (where defined) are even? odd?a. ƒg b.
Find the domains and ranges of ƒ, g, ƒ + g, and ƒ · g. f(x)=√x + 1, g(x)=√x - 1
Find the domain and range of each function. f(x) = 1 √x
Find the domain and range of each function. F(x) = √5x + 10
Find the domain and range of each function.ƒ(x) = 1 + x2
Find the domain and range of each function. f(t) 4 3-t
Find the domain and range of each function. g(x) = Vx² - 3x
Find the domain and range of each function. G(t) || 2 1² - 16
Find the domains and ranges of ƒ, g, ƒ/g, and g/ƒ.ƒ(x) = 1, g(x) = 1 + √x
If ƒ(x) = x + 5 and g(x) = x2 - 3, find the following.a. ƒ(g(0)) b. g(ƒ(0))c. ƒ(g(x)) d. g(ƒ(x))e. ƒ(ƒ(-5)) f. g(g(2))g. ƒ(ƒ(x)) h. g(g(x))
Which of the graphs are graphs of functions of x, and which are not? Give reasons for your answers.a.b. 0 2
Copy and complete the following table. a. g(x) 1 x-1 b.? c.? d. Vir f(x) 지 X 1 X Vx ? (fog) (x) ? X x + 1
Write a formula for ƒ ∘ g ∘ h. f(x) = x + 2 3 - x' g(x) = x² x² + 1 h(x) = √2-x V2
Evaluate each expression using the functionsa. ƒ(g(0)) b. g(ƒ(3)) c. g(g(-1))d. ƒ(ƒ(2)) e. g(ƒ(0)) f. ƒ(g(1/2)) f(x) = 2x, g(x) = -X₂ x - 1, -2≤x≤0 0≤x≤ 2.
If ƒ(x) = x - 1 and g(x) = 1/(x + 1), find the following.a. ƒ(g(1/2)) b. g(ƒ(1/2))c. ƒ(g(x)) d. g(ƒ(x))e. ƒ(ƒ(2)) f. g(g(2))g. ƒ(ƒ(x)) h. g(g(x))
Write a formula for ƒ ∘ g ∘ h.ƒ(x) = x + 1, g(x) = 3x, h(x) = 4 - x
Find the natural domain and graph the functions. g(x) = √-x
Write a formula for ƒ ∘ g ∘ h.ƒ(x) = 3x + 4, g(x) = 2x - 1, h(x) = x2
Express the side length of a square as a function of the length d of the square’s diagonal. Then express the area as a function of the diagonal length.
Let ƒ(x) = x - 3, g(x) = √x, h(x) = x3, and j(x) = 2x. Express each of the functions as a composite involving one or more of ƒ, g, h, and j.a. y = 2x - 3 b. y = x3/2c. y = x9 d. y = x -
The accompanying figure shows the graph of y = x2 shifted to two new positions. Write equations for the new graphs. - 0 5 Position (a) y = x² Position (b) X
Find the natural domain and graph the functions.ƒ(x) = 1 - 2x - x2
The accompanying figure shows the graph of y = -x2 shifted to four new positions. Write an equation for each new graph. (b) (-4,-1), (-2,3) (c) (1,4) (a) (2, 0) (d) > X
(a) Write formulas for ƒ ∘ g and g ∘ ƒ and find the (b) Domain(c) Range of each.ƒ(x) = x2, g(x) = 1 - √x
Graph the functions G(x) [1/x, Lx₂ x < 0 0≤x
Let ƒ(x) = 2x3 - 4. Find a function y = g(x) so that (ƒ ∘ g)(x) = x + 2.
Graph the functions g(x) = 1 - X, 2 - X, 0 ≤ x ≤ 1 1 < x < 2
Find the natural domain and graph the functions.G(t) = 1/|t|
Tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling
Graph the following equations and explain why they are not graphs of functions of x.a. |x| + |y| = 1 b. |x + y| = 1
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