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study help
mathematics
precalculus
Questions and Answers of
Precalculus
If you roll a 1-m-diameter wheel forward 30 cm over level ground, through what angle will the wheel turn? Answer in radians (to the nearest tenth) and degrees (to the nearest degree).
Which of the functions graphed are one-to-one, and which are not? y y = x¹/3
Graph the equation |x| + |y | = 1 + x.
What is meant by an increasing function? A decreasing function? Give an example of each.
Determine whether the graph of the function is symmetric about the y-axis, the origin, or neither.y = x2/5
Sketch the given curves together in the appropriate coordinate plane and label each curve with its equation.y = -ex and y = -e-x
Graph the equation y + |y| = x + |x|.
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
What is an even function? An odd function? What symmetry properties do the graphs of such functions have? What advantage can we take of this? Give an example of a function that is neither even nor
Determine from its graph if the function is one-toone. f(x) = √3-x, x
Determine whether the graph of the function is symmetric about the y-axis, the origin, or neither.y = x2 - 2x - 1
Determine from its graph if the function is one-toone. f(x) = 2x + 6, 2x + 4₂ x≤ −3 x > -3
If ƒ and g are real-valued functions, how are the domains of ƒ + g, ƒ - g, ƒg, and ƒ/g related to the domains of ƒ and g? Give examples.
Explain the following “proof without words” of the law of cosines. 2a cos - b a-c a C. a b OT
Determine whether the graph of the function is symmetric about the y-axis, the origin, or neither.y = e-x2
Sketch the shifted exponential curve.y = 3x + 2 and y = 3-x + 2
Find an appropriate graphing software viewing window for the given function and use it to display its graph. The window should give a picture of the overall behavior of the function. There is more
One of sin x, cos x, and tan x is given. Find the other two if x lies in the specified interval. tan x = 2, = xe|0. XE 0, TT 2
Determine from its graph if the function is one-toone. f(x) = 1 Xx 2' X .r + 21 r < 0 > 0
Show that the area of triangle ABC is given by (1/2)ab sin C = (1/2)bc sin A = (1/2)ca sin B. A b C C a B
When is it possible to compose one function with another? Give examples of composites and their values at various points. Does the order in which functions are composed ever matter?
Determine from its graph if the function is one-toone. f(x) = (2-x², x≤1 L.x², x > 1
One of sin x, cos x, and tan x is given. Find the other two if x lies in the specified interval. COS X 5 13' XE TT
How do you change the equation y = ƒ(x) to shift its graph vertically up or down by | k| units? Horizontally to the left or right? Give examples.
Sketch the shifted exponential curve.y = -1 - ex and y = -1 - e-x
Find an appropriate graphing software viewing window for the given function and use it to display its graph. The window should give a picture of the overall behavior of the function. There is more
Shows the graph of a function y = ƒ(x). Copy the graph and draw in the line y = x. Then use symmetry with respect to the line y = x to add the graph of ƒ-1 to your sketch. (It is not necessary to
Show that the area of triangle ABC is given by √s(s - a)(s - b)(s - c) where s = (a + b + c)/2 is the semiperimeter of the triangle.
How do you change the equation y = ƒ(x) to compress or stretch the graph by a factor c > 1? Reflect the graph across a coordinate axis? Give examples.
One of sin x, cos x, and tan x is given. Find the other two if x lies in the specified interval. sin x = -1/1 XE[T 2' 3π 2
Show that if ƒ is both even and odd, then ƒ(x) = 0 for every x in the domain of ƒ.
What is radian measure? How do you convert from radians to degrees? Degrees to radians?
Use the laws of exponents to simplify the expression.91/3 · 91/6
Use the laws of exponents to simplify the expression. 35/3 32/3
Find an appropriate graphing software viewing window for the given function and use it to display its graph. The window should give a picture of the overall behavior of the function. There is more
a. Let ƒ be a function whose domain is symmetric about the origin, that is, -x belongs to the domain whenever x does. Show that ƒ is the sum of an even function and an odd function: ƒ(x) = E(x) +
Graph the six basic trigonometric functions. What symmetries do the graphs have?
What happens to the graph of y = ax2 + bx + c asa. A changes while b and c remain fixed?b. B changes (a and c fixed, a ≠ 0)?c. C changes (a and b fixed, a ≠ 0)?
Shows the graph of a function y = ƒ(x). Copy the graph and draw in the line y = x. Then use symmetry with respect to the line y = x to add the graph of ƒ-1 to your sketch. (It is not necessary to
What is a periodic function? Give examples. What are the periods of the six basic trigonometric functions?
What is the period of each function?sin (x/2)
Find an appropriate graphing software viewing window for the given function and use it to display its graph. The window should give a picture of the overall behavior of the function. There is more
What happens to the graph of y = a(x + b)3 + c asa. A changes while b and c remain fixed?b. B changes (a and c fixed, a ≠ 0)?c. C changes (a and b fixed, a ≠ 0)?
An object’s center of mass moves at a constant velocity y along a straight line past the origin. The accompanying figure shows the coordinate system and the line of motion. The dots show positions
Use the laws of exponents to simplify the expression.(251/82)4
Shows the graph of a function y = ƒ(x). Copy the graph and draw in the line y = x. Then use symmetry with respect to the line y = x to add the graph of ƒ-1 to your sketch. (It is not necessary to
How does the formula for the general sine function ƒ(x) = Asin ((2π/B)(x - C)) + D relate to the shifting, stretching, compressing, and reflection of its graph? Give examples. Graph the general
Starting with the identity sin2 θ + cos2 θ = 1 and the formulas for cos (A + B) and sin (A + B), show how a variety of other trigonometric identities may be derived.
What is the period of each function?cos πx/2
Use the laws of exponents to simplify the expression.(13√2)√2/2
a. Find the slope of the line from the origin to the midpoint P of side AB in the triangle in the accompanying figure (a, b > 0).b. When is OP perpendicular to AB? y B(0, b) P A(a,0) X
Find an appropriate graphing software viewing window for the given function and use it to display its graph. The window should give a picture of the overall behavior of the function. There is more
Consider the quarter-circle of radius 1 and right triangles ABE and ACD given in the accompanying figure. Use standard area formulas to conclude that sin cos 0 < (0, 1) A 2 B E V 1 sin 0 2 cos
Name three issues that arise when functions are graphed using a calculator or computer with graphing software. Give examples.
Find an appropriate graphing software viewing window for the given function and use it to display its graph. The window should give a picture of the overall behavior of the function. There is more
Use the laws of exponents to simplify the expression.2√3 · 7√3
What is an exponential function? Give examples. What laws of exponents does it obey? How does it differ from a simple power function like ƒ(x) = xn? What kind of real-world phenomena are modeled by
Use the laws of exponents to simplify the expression. 2 2 4
What is the period of each function?-cos 2πx
If ƒ(a - x) = ƒ(a + x), show that g(x) = ƒ(x + a) is an even function.
Use the laws of exponents to simplify the expression.(√3)1/2 · (√12)1/2
a. Graph the function ƒ(x) = 1/x. What symmetry does the graph have?b. Show that ƒ is its own inverse.
What is the period of each function? sin(x + TT 6
Let ƒ(x) = ax + b and g(x) = cx + d. What condition must be satisfied by the constants a, b, c, d in order that (ƒ ∘ g)(x) = (g ∘ ƒ)(x) for every value of x?
Find an appropriate graphing software viewing window for the given function and use it to display its graph. The window should give a picture of the overall behavior of the function. There is more
What is the number e, and how is it defined? What are the domain and range of ƒ(x) = ex? What does its graph look like? How do the values of ex relate to x2, x3, and so on?
Find the (a) Domain and (b) Range. y = -2 + V1 - x
Gives a formula for a function y = ƒ(x) and shows the graphs of ƒ and ƒ -1. Find a formula for ƒ -1 in each case. f(x) = x², x ≤0 y = f(x) 1 y 0 1 y=f-¹(x) X
Use the laws of exponents to simplify the expression. 3 2
Leta. Give a convincing argument that ƒ is one-to-one.b. Find a formula for the inverse of ƒ. f(x) = ax + b cx + d' c = 0, ad - - bc # 0.
What functions have inverses? How do you know if two functions ƒ and g are inverses of one another? Give examples of functions that are (are not) inverses of one another.
What is the period of each function? cos | x + 27 3 1 2
Gives a formula for a function y = ƒ(x) and shows the graphs of ƒ and ƒ -1. Find a formula for ƒ -1 in each case. - 0 y f(x) = x² − 2x + 1, x ≥ 1 y = f-¹(x) 1 y = f(x) X
How are the domains, ranges, and graphs of functions and their inverses related? Give an example.
What procedure can you sometimes use to express the inverse of a function of x as a function of x?
Find an appropriate graphing software viewing window for the given function and use it to display its graph. The window should give a picture of the overall behavior of the function. There is more
What is a logarithmic function? What properties does it satisfy? What is the natural logarithm function? What are the domain and range of y = ln x? What does its graph look like?
Find the (a) Domain and (b) Range.y = 32-x + 1
Find the domain and range for each of the function.g(t) = cos (e-t)
Find an appropriate graphing software viewing window for the given function and use it to display its graph. The window should give a picture of the overall behavior of the function. There is more
Find the domain and range for each of the function. f(x) || 3 1 - e2x
How is the graph of loga x related to the graph of ln x? What truth is in the statement that there is really only one exponential function and one logarithmic function?
How are the inverse trigonometric functions defined? How can you sometimes use right triangles to find values of these functions? Give examples.
Gives a formula for a function y = ƒ(x) and shows the graphs of ƒ and ƒ -1. Find a formula for ƒ -1 in each case. f(x) = x²/3, x ≥ 0 y 0 y = f¯¹(x) 1 y = f(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 = -tan πt
Find the (a) Domain and (b) Range.y = tan (2x - π)
Graph the functions. The ts-plane (t-axis horizontal, s-axis vertical). What is the period of each function? What symmetries do the graphs have? S = CSC 2
If you use the approximation ln 2 ≈ 0.70 (in place of 0.69314c), you can derive a rule of thumb that says, “To estimate how many years it will take an amount of money to double when invested at r
For what x > 0 does x(xx) = (xx)x? Give reasons for your answer.
a. If (ln x)/x = (ln 2)/2, must x = 2?b. If (ln x)/x = -2 ln 2, must x = 1/2?Give reasons for your answers.
Find the (a) Domain and (b) Range.y = x2/5
Use graphs to find approximate solution.ex = 4
Find an appropriate graphing software viewing window for the given function and use it to display its graph. The window should give a picture of the overall behavior of the function. There is more
a. Graph y = cos x and y = sec x together for -3π/2 ≤ x ≤ 3π/2. Comment on the behavior of sec x in relation to the signs and values of cos x.b. Graph y = sin x and y = csc x together for -π
Find an appropriate graphing software viewing window for the given function and use it to display its graph. The window should give a picture of the overall behavior of the function. There is more
Gives a formula for a function y = ƒ(x). In each case, find ƒ -1(x) and identify the domain and range of ƒ -1. As a check, show that ƒ(ƒ -1(x)) = ƒ -1(ƒ(x)) = x.ƒ(x) = x4,
Find the (a) Domain and (b) Range. 3, y = −1+√³/2 - x
The quotient (log4 x)/(log2 x) has a constant value. What value? Give reasons for your answer.
How does ƒ(x) = logx (2) compare with g(x) = log2 (x)? Here is one way to find out.a. Use the equation loga b = (ln b)/(ln a) to express ƒ(x) and g(x) in terms of natural logarithms.b. Graph ƒ and
Find the largest interval on which the given function is increasing.a.b.c.d. f(x) = |x - 2 + 1
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