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Precalculus final study
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Terms in this set (63)
Consider the inverse function.
f^-1(x)= -\|x-2
Which conclusions can be drawn about f(x) = x2 + 2? Select three options.
f(x) has a limited range.
f(x) has a restricted domain.
f(x) has a y-intercept at the point (0, 2).
The inequality 10x ≤ -0.6x + 7 can be broken into two related inequalities. Review the graphs of the two related inequalities. What is the least integer value in the solution set of the inequality?
-3
For the function q(x) =10x ^5\|x-2, what is the inverse function?
q-1(x) =(x/10)^5+2
Randy and Trey take turns cleaning offices on the weekends. It takes Randy at most 4 hours to clean the offices. It takes Trey at most 6 hours to clean the offices. What is the greatest amount of time it would take them to clean the offices together?
2.4 hours
Consider the function represented in the table. Which point of the given function corresponds with the minimum value of its inverse function?
A (-20, 8)
Suppose f(x) = (5 - x)3 and, g(x) = a + x2, and f(g(x)) = (1 - x2)3. What is the value of a?
a = 4
One brand of vinegar has a pH of 4.5. Another brand has a pH of 5.0. The equation for the pH of a substance is pH = -log[H+], where H+ is the concentration of hydrogen ions. What is the approximate difference in the concentration of hydrogen ions between the two brands of vinegar?
2.2x10^-5
Which statement comparing the oblique asymptotes of the functions f(x) and g(x) is true?
f(x)=x^2+x+1/6(x-4)
The oblique asymptote for g(x) is steeper than for f(x).
Both functions have the same oblique asymptote.
The oblique asymptote for f(x) is steeper than for g(x).
Both functions have parallel oblique asymptotes.
The oblique asymptote for f(x) is steeper than for g(x).
If the interval (a, ∞) describes all values of x for which the graph of is decreasing, what is the value of a?
3
The function h(x) = -x5 + 12x is an odd function. Which transformation of h(x) results in an odd function?
-6h(x)
Two different radioactive isotopes decay to 10% of their respective original amounts. Isotope A does this in 33 days, while isotope B does this in 43 days. What is the approximate difference in the half-lives of the isotopes?
3 days
Consider the function.
f(x) = x2 + 3
Which answer pairs a possible domain restriction for f(x) and its corresponding impact on f-1(x)?
f(x) domain: x ≥ 0
f-1(x) range: y ≥ 0
How can the graphs of f(x) =5/4(x-28) and g(x) =4/5x + 28 be used to prove that f(x) and g(x) are inverse functions?
Show that the graph of g(x) is a reflection about the line y = x of the graph of f(x).
Review the table of values for function w(x).
What is the value of w^-1(3)?
-0.75
Function 1:
f(x)=x^2+3x-40/x-4
Function 2:
g(x)=x^2+4x-32/x-4
Which statements comparing the graphs of function 1 and function 2 are true? Check all that apply.
Function 1 has an oblique asymptote, while function 2 does not.
Function 2 has a hole, while function 1 only has asymptotes.
Both functions have the same domain restriction.
Judy knows that the function graphed below is a transformation of the parent function . Since the period and asymptotes are the same as those of the parent function, she knows that there were no horizontal stretches, compressions, or translations. She can also tell that there were no reflections since the general shape of the graph is the same as that of the parent function.
C. Since the y-intercept of the graph is (0,-1.5) , the graph of the parent function was translated 1.5 units down, and since the graph passes through the point (pi/4,-1) instead of (pi/4,-0.5) , the graph was vertically compressed.
It takes 1.5 seconds for a grandfather clock's pendulum to swing from left (initial position) to right, covering a horizontal distance of 10 inches.
Which function models the horizontal displacement as a function of time in seconds?
y=5cos(2pi/3x)
A disc spins at a rate of 6.4 revolutions per minute. To the nearest tenth, what is the linear velocity, in meters per minute, of a point on the disc that is 3 cm from the center of the disc?
1.2 m/min
Which is equal to -214°?
-107pi/90 radians
Which graph represents y=cos(2x-pi)+5?
A
Which point does the graph of the parent function y=tan(x) pass through?
(pi/3,rad3)
On which interval are both the sine and cosine functions increasing?
(3pi/2,2pi)
What restriction should be applied to y = tanx for y = arctanx to be defined?
restrict the domain (-pi/2, pi/2)
Let cot(x) = 5/8 , with 0 < x < 90°. What is sin(x)?
8/rad89
The average high temperature during the hottest month in Beijing, China, is 72°F. The average high temperature during the coldest month in Beijing is 18°F. The function , where m represents the number of months after January, models the temperature throughout the year in Beijing.
What is the first month in the year when the temperature reaches 55°F?
April
A waterwheel has a radius of 5 feet. The center of the wheel is 2 feet above the waterline. You notice a white mark at the top of the wheel. How many radians would the wheel have to rotate for the white mark to be 3 feet below the waterline?
pi radians
Consider the graphs of j(x) = and k(x) = .
Which feature of the graph of k(x) is different from the graph of j(x)?
the vertical shift
Review the graph of y = sin(x). Based on the graph, what are the domain and range of the sine function?
domain: (-ꝏ, ꝏ); range: [-1, 1]
What is the value of cos(tan-1(0))?
1
A circular crop field has an irrigation system in which a pipe, with one end attached to a tower at the center of the field, turns continuously to deliver water. The horizontal distance, in meters, from the tip of the pipe to the farmhouse is represented by y =400sin(pi/8x+pi)+500 , where x represents time in hours.
the length of the pipe
The motion of a weight that hangs from a spring is represented by the equation h=-5sin(3pi/4t). It models the weight's height, h, in inches above or below the rest position as a function of time, t, in seconds. Approximately when will the object be 4 inches below the rest position? Round to the nearest hundredth, if necessary.
0.39 seconds
To solve the trigonometric inequality over the interval radians, Gabby performed the following steps, first simplifying the inequality and then setting both sides equal to y and graphing the equations.
In which step did she first make an error, and what was the error?
C.She first made an error in step 5 because she did not correctly graph y=cot(x)-csc(x).
Which function is equivalent to the inverse of ?
y=cos^-1(1/x)
Let sin(2x) = cos(x), where 0° ≤ x < 180°. What are the possible values for x?
30°, 90°, or 150°
What is the exact value of tan(19pi/12)?
-2-\|3
For what values of x is the trigonometric equation tan(x − 180°) − sin(x − 180°) = 0, where 0 ≤ x < 360°, true?
{0°, 180°}
What is the value of 1 − 2sin2(105°)?
-\|3/2
Which statement is true about the graph of the equation y=csc^-1(x)?
There is a horizontal asymptote at y=0?
Which shows a correct simplification of sin(x + )?
sin(x + pi)
= sin(x)cos(pi) + cos(x)sin(pi)
= sin(x) · -1 + cos(x) · 0
= -sin(x)
For 0 ≤ x < 2π, which of the following solves 2cos(x/2) + 1 = 0?
{4pi/3}
What is the exact value of sin(105°)?
rad2+rad3/2
The following statements are about finding a midpoint of a segment in the complex plane. Which statement is false?
First subtract the endpoints and then divide by 2.
Which region on the unit circle contains a cube root of - 8 - 8i?
in quadrant I
Which property of multiplication is shown below?
If x = 3 + 4i and y = 5 + 6i, x × y = y × x.
commutative property
Review the proof of de Moivre's theorem.
What is the justification for step B?
assumption (for n = k step)
The sum of two complex numbers, where the real numbers do not equal zero, results in a sum of 22i. Which statement must be true about the complex numbers?
The complex numbers have opposite real numbers.
Review the graph.
Given a pair of complex numbers, z1 = 5 and z2 = -6i, let z3 = 3i + z1 and z4 = z2 - 2. In which quadrant is z3 - z4 located?
quadrant I
Consider the polar curve r = 4sin(2θ). What is the length of each petal, and how many petals does the graph have?
The length of each petal is 4, and the number of petals is 4.
Which expression is equivalent to (3 + 5i)2?
-16 + 30i
An isosceles triangle's altitude will bisect its base.
Which expression could be used to find the area of the isosceles triangle above?
rad40xrad40/2
The radar system in an airport determines the location of inbound and outbound planes once they are within range. The range, based on the signal from the tower, is represented by the polar equation r = 10cos(θ). Which graph represents the signal range of the airport's radar system?
A.
Given z1 = 2 + rad3i and z2 = 1 - rad3i , what is the sum of z1 and z2?
3
Identify the horizontal and vertical asymptotes, if any, for the graph of f(x) shown. Check all that apply.
y = 0, x=1
Suppose f(x)= 2x-1 and g(x)=x+1/a. Which value(s) of a would make the composite functions commutative?
2
The time it takes a spinner to stop completely is 12 seconds. If the spinner completes 6 revolutions in this time, what was the average angular velocity of the spinner, in radians per second, for the 12-second interval?
pi
Which values of x and y would make the following expression represent a real number?
x = 6, y = -3
What is the domain of the function y=tan(x/8)?
all real numbers except 4pi+8npi, where is any integer
A vector in standard position has its terminal point at (7, -4). What is the magnitude of the vector?
rad65
The parent cosecant function is shifted 2 units down, and its period is changed to 6pi. Which of the following is the graph of the transformed function?
A.
Consider the function f(x) = 10(x2 - 7x). What are the additive and multiplicative inverses?
B. The additive inverse is g(x) = -10(x2 - 7x) and the multiplicative inverse is h(x) = 1/(10(x^2-7x)).
Which region on the unit circle contains a cube root of - 8 - 8i?
in quadrant I
The following statements concern the modulus. Which statement is false?
The modulus is sometimes negative because of the computations with i.
Two different cars each depreciate to 60% of their respective original values. The first car depreciates at an annual rate of 10%. The second car depreciates at an annual rate of 15%. What is the approximate difference in the ages of the two cars?
1.7 years
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