Precalculus Review Part II: Inverse Functions, Exponential, Logarithmic, and Trigonometric Functions 1. The graph of f is given. (a) Why is f one-to-one? (b) What are the domain and range of f −1 ? (c) What is the value of f −1 (2) ? (d) Estimate the value of f −1 (0) ?

2. Given the graph of y = e x , write a separate equation for each of the graphs that would result from performing the given translation or transformation. (a) shifting the graph 1 units to the right and 2 units downward. (b) reflecting the graph about the x-axis. (c) reflecting the graph about the line y = x. 3. Determine the values of a and C for the function y = Ca x if the graph of f passes through the points

3   1 −1  9   −1,  and 1,  . Find f   . 2   6 2 4. Solve for x , leaving your answer in exact form: (a)

e −2ln 5

(b)

2 < ln x < 9

5. Solve the following logarithmic equations. (a)

4 + ln 3x = 2

(b) log 10 x + log 10 ( x + 3) = 1

6. Determine if the following pairs of parametric equations describe the same curve, including orientation. If not, how do they differ?

(a)

x = t2, y =1− t2, 0 ≤ t ≤ 1

(b)

x = 1 − ln t , y = ln t , 1 ≤ t ≤ e

2 x cos= t , y sin 2 t , 0 ≤ t ≤ 2π (c) =

7. The number of bacteria in a certain culture increases from 5,000 to 15,000 in 10 hours. Assuming that the rate of increase is proportional to the amount present, find two different formulas for the number of bacteria in the culture at any time, t. How many bacteria are present in the culture at the end of 20 hours? When will the number be 86,000? 8. Match the functions x(t) and y(t) in parts a)-d) with the graphs I-IV

9. Without using a calculator, determine the exact values of the six trigonometric functions of the following special angles. a) 60°

b) 135°

c) 180°

d) 210°

10. Give the reference angle , then a positive angle and a negative angle coterminal with each of the following: a) −210°

b) 315°

11. An angle θ is in standard position on a rectangular coordinate system. Find the quadrant that θ lies in if: a) sec θ < 0 and sin θ > 0 b) cot θ > 0 and cscθ > 0 12. Determine the values of the six trigonometric functions of θ , assuming that the terminal side of θ lies in quadrant III on the line x - 3y = 0. 13. An airplane takes off at an angle of 10 degrees and travels at the rate of 250 ft/sec. Approximately how long does the airplane take to reach an altitude of 28,000 feet? 14. Find all solutions for the following equations. a) sin 2 x =

1 2

b) cos 2 x + cos x = 0

c) sin 2 x + cos x = 0

15. A prop plane leaves the airport traveling at 215 mph at a heading of 65. 4° at the same time a jet plane leaves the airport traveling at 480 mph at a heading of 335. 4° . Find the distance between them after two hours and the bearing of the prop plane from the jet plane. 16. Evaluate the following expression without using a calculator.

sin 2 210°+ cos 270°− sec 2 120°+ tan 225° 17. Verify the following identity (work on one side of the identity only)

  3π cos − β  = − sin β  2  18. Verify the following identity (work on one side of the identity only)

cosθ + tan θ = sec θ 1+ sin θ

19. Solve: a) tan 2 α − 2 sec α = 2 b) tan 2 x =

for α ∈ [ 0,2π )

3 tan x for x∈ [ 0° ,360°) .

20. Find the exact value of the following:

 1  2

  

a) arccos − 

b) tan  sin −1 −

2  2 

21. If a projectile is fired from ground level with an initial velocity of v ft/sec and at an angle of θ degrees with the horizontal, the range R of the projectile is given by

R=

v2 sin θ cosθ . 16

If v = 80 ft/sec, approximate the angles that result in a range of 150 feet. 22. Solve for x :

 3 π arccos( − x ) + arcsin  = .  2  2

23. Find two possible equations for the graph below.

24. For several hundred years astronomers have kept track of the number of solar flares, or sunspots, that occur on the surface of the sun. The number of sunspots counted in a given year varies periodically from a minimum of about 10 per year to a maximum of about 110 per year. Between the maximums that occurred in the years 1750 and 1981, there were 21 complete cycles. a) What is the period of the sunspot cycle? b) Assume that the number of sunspots counted in a year varies sinusoidally with the time in years. Sketch a graph of 2 sunspot cycles, beginning with 1981. c) Write an equation expressing the number of sunspots per year (y) in terms of the year (t). d) How many sunspots would you expect this year? In the year 2020? e) What is the first year after 2000 in which the maximum number of sunspots will occur?

25. A graph of y = cos 2 x + 2 cos x for 0 ≤ x ≤ 2π is shown below. The x- coordinates of the turning points P, Q and R on the graph are solutions of the equation sin 2 x + sin x = 0 . Find the exact values of the coordinates of these points.