Modern States Pre-Calculus

Based on past sales, a convenience store has observed a linear relationship between the number of units of Product X that will be sold to customers each week and the price per unit. The figure above models this linear relationship. Based on the model, how many dollars would the convenience store expect to earn from its sales of Product X in a week when the price per unit is $5 ?

$350

Suppose f(x)=x^2−4, g(x)=ln⁡(x). What is the domain of (g∘f)(x)?

(-∞,-2) ∪ (∞,2)

What is the range of −(x+2)^2+4?

(-∞,4)

If log base 2b x^5=T and b>1/2, then x=

(2b)^T/5

Find the solution (x,y,z) to the following linear system x+y=2 2x−3y+z=1 y-z=0

(5/4, 3/4, 3/4)

A portion of the graph of a function f is shown above. The domain of f is the set of all real numbers. Which of the following could be the equation of f ?

(f(x)=3sin⁡(2x−π)

What is the domain of the function y=log⁡(tan⁡x)?

(kπ,kπ+π/2) for all integer k

The graph in the xy-plane of which of the following equations is a circle?

(x−3)^2=25−(y+2)^2

The circle with center P is tangent to perpendicular lines l and k at points (1, 3) and (3, -1), respectively, as shown above. Which of the following is an equation of the circle with center P ?

(x−4)^2+(y−2)^2=10

If (x−√5)(x+√5)=5, what is the value of x ?

+ and - √10

f(x)=x(x+1) g(x)=x^2−1 The functions f and g are defined above. If f(a)=g(a), what is the value of a ?

-1

For all x≠0, the function f is defined by f(x)=x|x|. What is the range of f?

-1 and 1 only

Solve the equation 2x^2+3x−4=x^2+4x−2.

-1, 2

g(x) = 1/x and h is a nonzero real number, then (g(x + h) - g(x))/h =

-1/(x(x+h))

In the xy-plane, the vertex of the parabola x=y^2+4y+1 is the point (h,k). What is the value of k ?

-2

The graph of the function f and a table of values for the function g are shown above. What is the value of f(g(0))?

-2

x−y=1 x^2+y^2=5 The point (x,y) lies in the third quadrant of the xy-plane and satisfies the equations above. What is the value of y ?

-2

In the xy-plane, the lines with equations 2x+2y=1 and 4x−y=4 intersect at the point with coordinates (a,b). What is the value of b ?

-2/5

Let f be the function defined by f(x)=5sin⁡(2x)+1 for 0≤x≤π. What is the slope of the line passing through the maximum and minimum points of the function on the interval?

-20/π

If 4=(t−2√3)(t+2√3), which of the following could be the value of t?

-4

Let f(x)=2x,g(x)=x−1,h(x)=x^2. Compute (f∘g∘h) (-1)

0

The figure above shows the graph of the function f defined by f(x)=2x+4. If f^−1 is the inverse function of f , what is the value of f^−1 (2)?

0

In the figure above, line l passes through the origin and intersects the graph of y=2^−x at the point (a,0.4). What is the slope of line l?

0.303

4cos⁡x=9sin⁡x Which of the following is the solution to the equation above in the interval [0,π/2] ?

0.4182

The graph of y=a(b^x), where a and b are constants and b>0, is shown above. If the points (0,−1) and (2,−0.25) are on the graph, what is the value of b ?

0.5

Compute tan arcsin (√2/2)

1

Let f be the function defined by f(x)=−|x|. The graph of the function g in the xy-plane is obtained by first translating the graph of f horizontally 3 units to the left and then vertically translating this result 2 units up. What is the value of g(-2) ?

1

Simplify tan⁡(x)cos⁡(x)csc⁡(x).

1

Suppose f(x)=(x^5)−1. Compute f^−1(0).

1

(sin t + cos t)^2

1+sin2t

Solve the equation sec(θ)=5, for 0≤θ≤π/2.

1.369

For all x such that 0<x<π/2, which of the following is equivalent to sin⁡(2x)sin⁡x − cos⁡(2x)cos⁡x?

1/cosx

Suppose a triangle with angles 40°, 60°, 80° has corresponding side lengths 4,x,y. Compute x+y.

11.518

Suppose a population of mice starts with 10, and doubles every 3 days. How long until there are 200 mice, rounded to the nearest thousandth?

12.966

What is the x-intercept of the graph of y = 1/8x ^ (3/2) - 8

16

Suppose a colony of bacteria has membership at time t given by P(t)=100∗2^t/50. How long until there are 1000 members of the colony?

166.0964

The function f defined by f(x)=−x^2 is graphed in the xy-plane above. The graph of the function g is obtained by reflecting the graph of f across the line y=1. What is the value of g(4) ?

18

A transport authority plans to construct a bridge between City A and City B. To determine the distance between the cities, a surveyor team uses the triangular region shown. What is the distance between City A and City B, to the nearest tenth of a mile?

2.6 miles

If a and b are numbers such that ln⁡a=2.1 and ln⁡b=1.4, what is the value of ln⁡(a^2/b)?

2.8

Suppose f(x) is a quadratic function with f(0)=3, f(1)=0, f(2)=3, f(3)=12. Compute f(4).

27

Simplify the following expression sin⁡(2θ)sec⁡(θ).

2sin(θ)

The value of log(1732) is between what two integers?

3 and 4

What is the area of the following ellipse: x^2/25 + y^2/49 = 1

35π

How many solutions to tan⁡(2θ)=√3 are there in the interval 0≤θ≤2π?

4

Simplify (2−√t)(2+√t) for t>0

4-t

If π≤θ≤2π and cos⁡θ=cos⁡1, what is the value of θ ? Round your answer to the nearest hundredth.

5.28

Let f(x)=(2x^5)+3. Compute f^−1(x).

5th root of 1/2x-3/2

The measure of a certain angle is 25°. What is the corresponding radian measure of the angle?

5π/36

The table above shows some values for the function f . If f is a linear function, what is the value of a+b ?

64

A(t)=ke^−0.001t, where k is a constant. When a certain radioactive element decays, the amount, in milligrams, that remains after t years can be approximated by the function A above. Approximately how many years would it take for an initial amount of 800 milligrams of this element to decay to 400 milligrams?

693

If f(x)=2x+1 and g(x)=3x−1, then f(g(x))=

6x-1

In the xy-plane, the graph of y=x^2+bx+c is symmetric about the line x=3 and passes through the point (5,2). What is the value of c ?

7

The population of city Z was 420,000 in the year 2000. If the population is projected to grow at a constant rate of 2 percent per year, which of the following is closest to the projected population of city Z in the year 2030?

760,000

Find the solution to log base 3⁡(x+2)=4, to the nearest thousandth.

79.000

Ellen put $1,000 into a new savings account that paid 5% interest per year, compounded continuously. She made no deposits to or withdrawals from the account, and the amount, A(t), in dollars, in the account t years from the time she opened the account is given by the function above. At this rate, how many years from the time she opened the account did the value of the account increase by 50 percent? (Round your answer to the nearest tenth of a year. Type your answer into the box below.)

8.1

An experiment designed to measure the growth of bacteria began at 2:00 p.m. and ended at 8:00 p.m. on the same day. The number of bacteria is given by the function N, where N(t)=1000⋅3^2t/3 and t represents the number of hours that have elapsed since the experiment began. How many more bacteria were there at the end of the experiment than at the beginning of the experiment?

80000

The table above shows selected values for the function p . If p is a quadratic polynomial, what is the value of p(10) ?

91

How many different values of x satisfy the equation sin⁡x+2sin⁡(2x)=√x?

Five

Which of the following relations define y as a function of x ?

II and III

h(x)=(x^2 e^x)/x The function h is defined above. Which of the following are true about the graph of y=h(x)?

II and III only

A rectangular box with a square base is open at the top and has a volume of 12 cubic feet. Each side of the base has a length of feet Which of the following expresses the surface area, S. in square feet, of the outside of the box in terms of x?

S= x^2 + 48/x

The figure above shows the graph of a polynomial function g. Which of the following could define g(x)?

g(x)=-x^3+4x

A ball is dropped from an initial height of d feet above the floor and repeatedly bounces off the floor. Each time the ball hits the floor, it rebounds to a maximum height that is 3/4 of the height from which it previously fell. The function h models the maximum height, in feet, to which the ball rebounds on the nth bounce. Which of the following is an expression for h(n)?

h(n)=((3/4)^n)d

What are all solutions to the equation e^2x−e^x−2=0?

ln2 only

The population P of fish, in thousands, in a certain pond at time t years is modeled by the function P(t)=1/(1+(1/P0−1)e−rt), where P0 is the population at time t=0 and r is the growth rate of the population. If P(1)=5, which of the following is equivalent to r?

ln⁡(5(P0−1)/4P0)

Which of the following equations has no real solutions?

sin(2x+1)=2

In the xy-plane, the graph of y=x(x^2−2)(x^2+x+1) intersects the x-axis in how many different points?

three

Which of the following does not define y as a function of x?

x-3y^2=0

Find the solution to e^2x=cos⁡(x) with −2≤x≤−1, rounded to the nearest thousandth.

x=-1.523

Find the solutions of x^2−4x+1=0 to the nearest thousandth.

x=0.268, 3.732

f(x)=e^(x/4) g(x)=2sin⁡x The functions f and g are defined above and their domains are all real numbers. On what interval is the value of f(x) greater than the maximum value of g ?

x>2.773

Which of the following curves is not an ellipse?

x^2 = 1+y^2 correct

Which of the following could be an equation of the function graphed in the xy-plane above?

y=2sin⁡(3/2x)

If 0<θ<π/2 and 10sin⁡θ=z, what is tan⁡θ in terms of z?

z/(√100-z^2)

The Statue of Liberty is 46 meters tall and stands on a pedestal that is 47 meters above the ground. An observer is located d meters from the pedestal and is standing level with the base, as shown in the figure above. Which of the following best expresses the angle θ in terms of d ?

θ=arctan⁡(93/d)−arctan⁡(47/d)

In the xy-plane, which of the following is an equation of a vertical asymptote to the graph of y=sec⁡(6x−π)?

π/4

tan^-1(2cosπ/3)

π/4

What are all solutions of the equation cos⁡(2x)+1=sin⁡(2x) in the interval [0,2p)?

π/4,π/2,5π/4, and 3π/2

Let the function h be defined by h(x)=2cos⁡(10x)+12. The maximum value of h is attained at which of the following values of x ?

π/5

The function h is given by h(x)=log base 2 (x^2+2). For what positive value of x does h(x)=3?

√6

Let f and g be the functions defined by f(x)=x+2 and g(x)=x^2−a, where a is a positive constant. What are all values of a for which the graphs of f and g have exactly one point of intersection?

a>4 only

The equation of the line shown in the graph above is y=ax+b. Which of the following is always true for this line?

ab<0

The function f is given by f(x)=x+|x−10|. Which of the following defines f(x) for all x≤10?

f(x)=10

For each of the following functions, indicate which is even.

f(x)=9e^x + e−^x)/2 g(x)= |sinx|

Which of the following functions exhibits a horizontal asymptote?

f(x)=arctan⁡(x)

Which of the following functions has period 4π, when expressed in radians?

f(x)=cos(1/2x)

Which of the following functions exhibits neither even nor odd symmetry?

f(x)=sin⁡(x)+cos⁡(x)

The figure above shows the graph of a polynomial function f . What is the least possible degree of f?

five