14 Chapter 1: Functions

Set 1: Multiple-Choice Questions on Functions: Reviewt

Directions: Answer these questions without using your calculator.

1. If f(x) = x3-2x-l, then/(-2) =

(A) -17 (B) -13 (C) -5 (D) -1 (E) 3 X - 1

2. The domain of f(x) = -=—- is J X2 + 1

(A) all x * 1 (B) all x* 1,-1 (C) alljc*-l (D) JC ^ 1 (E) all reals

3. The domain of g(x) = —= is X — X

(A) a l l**0 , 1 (B) x ^ 2 , ; c * 0 , 1 (C) x^2 (D) x^2 (E) x > 2

4. Iff(x) = x3 - 3JC2 - 2x + 5 and g(jc) = 2, then g(/(jc)) =

(A) 2x3 - 6x2 - 2JC + 10 (B) 2JC2-6JC+1 (C) -6 (D) -3 (E) 2

5. With the functions and choices as in Question 4, which choice is correct for/*(g(;t))?

6. If/(x) = JC3 + Ax2 + Bx - 3 and if/(l) = 4 and/(-l) = -6, what is the value of 2A + £?

(A) 12 (B) 8 (C) 0 (D) -2 (E) It cannot be determined from the given information.

7. Which of the following equations has a graph that is symmetric with respect to the origin?

x - 1 (A) y= (B) y = 2x4+l (C) y = x' + 2x

x (D) v = x3 + 2 (E) y= j 3 + [

8. Let g be a function defined for all reals. Which of the following conditions is not sufficient to guarantee that g has an inverse function? (A) gix) = ax + b, a ̂ 0. (B) g is strictly decreasing. (C) g is symmetric to the origin. (D) g is strictly increasing. (E) g is one-to-one.

tBecause students in AP courses are assumed to have firm control of precalculus topics, such topics are not directly tested on the AP examinations. Nevertheless, some of this chapter's multiple-choice questions deal with those topics to reinforce important basic principles.

Set 1: Multiple-Choice Questions on Functions: Review 15

9. Let y =/(JC) = sin (arctan JC). Then the range of / i s

(A) { y | 0 < y = i l } (B) [y\-Ky<l) (C) { y | - l s i y : g l } i 7t _

(D) yi -7C ft] —<y<—> 2 ' 2J

(E) y — ^sy =s • r | 2 i

10. Let g(x) = |cos x - 11. The maximum value attained by g on the closed interval [0, 2ri\ is for JC equal to

(A) -1 (B) 0 (O § (D) 2 (E) n

11. Which of the following functions is not odd?

(A) /(jc) = sinjc (B) /(jc) = sin2jc (C) f(x) = x3+l

0>) /(*) = J C 2 + 1

(E) /(*)= nx

12. The roots of the equation f(x) = 0 are 1 and -2. The roots of/(2JC) = 0 are

(A) 1 and -2 (B) - and -1 (C) - - and 1

(D) 2 and-4 (E) -2 and 4

13. The set of zeros of f(x) = x? + Ax1 + 4x is

(A) {-2} (B) {0,-2} (C) {0,2} (D) {2} (E) {2,-2}

14. The values of x for which the graphs of y = x + 2 and y2 = Ax intersect are

(A) -2 and 2 (B) -2 (C) 2 (D) 0 (E) none of these

15. The function whose graph is a reflection in the y-axis of the graph of fix) = 1 - 3* is

(A) g(*) = l - 3 - (B) g(*) = l + 3 * (C) £(*) = 3*-l (D) g(*) = log3(Jc-l) (E) gOc) = log 3 ( l -*)

16. Let/(jc) have an inverse function g(x). Then/(g(;c)) =

1 (A) 1 (B) JC (C) (D) /(JC) • g(jc) (E) none of these

17. The function/(JC) = 2JC3 + JC - 5 has exactly one real zero. It is between

(A) - 2 a n d - l (B) - landO (C) Oandl (D) 1 and 2 • (E) 2 and 3

16 Chapter 1: Functions

In 18. The period of f(x) = sin — x is

(A) i (B) | (C) | (D) 3 (E) 6

19. The range of y =/(:*;) = In (cos JC) is

(A) { y l - o o ^ ^ O } (B) { y | 0 < y ^ l } (C) { y | - l < y < l } K 7t - < y < — 2 2

(D) ^ | - ^ < y < ^ (E) { y | 0 = i y = i l }

20. If log, (3*)= | , thenfc =

(A) i (B) i (C) i (D) 3 (E) 9

21. Let/-1 be the inverse function of/(jc) = x3 +2. Then/~'(;c) =

x'-2 (A) -r-; (B) (x + 2)3 (C) (JC-2) 3

(D) 4x~+2 (E) 4x^1

22. The set of x-intercepts of the graph of f(x) = ;c3-2j<:2-jc + 2is

(A) {1} (B) {-1,1} (C) {1,2} (D) {-1,1,2} (E) {-1,-2,2}

23. If the domain of/is restricted to the open interval (-^, ^ ) , then the range of f(x) = etmx is

(A)' the set of all reals (B) the set of positive reals (C) the set of nonnegative reals (D) {y | 0 < y ^ 1} (E) none of these

24. Which of the following is a reflection of the graph of y =/(*) in the *-axis?

(A) y = -/(*) (B) y =/(-*) (C) y = \f(x)\ (D) y=/(W) (E) y = -/(-*)

25. The smallest positive x for which the function/(JC) = sin ( |) - 1 is a maximum is

(A) | (B) n (C) y (D) 37t (E) 6n

Set 1: Multiple-Choice Questions on Functions: Review 17

26. tan (arccos (~pj) =

(A) -1 (B) ' - £ 3

(C) - (D) | (E) 1

27. If /-*(JC) is the inverse of/(jc) = 2e~\ thenf-'ix) =

(A) l„g) (B) tag) (C) g ) (D) Vm~c (E) ln(2-jc)

hue

28. Which of the following functions does not have an inverse function?

ft —• —-ft (A) y = sinx [ - - = § * ^ - (B) y = x» + 2 (C) v = JC2+1

(D) v= i** (E) y = ln(jc-2) (where A; > 2)

29. Suppose that/(jc) = In JC for all positive JC and g(;t) = 9 - JC2 for all real JC. The domain of f(g(x)) is

(A) { J C | J C ^ 3 } (B) { JC | | JC |^3 } (C) {JC||JC|>3} (D) {JC |JC|<3) (E) { J C | 0 < J C < 3 }

30. Suppose (as in Question 29) that/(jc) = In JC for all positive JC and g(x) = 9 - JC2 for all real JC. The range of v =/(g(jc)) is

(A) { v | v > 0 } (B) {y |0<y;£ln-9} (C) { v | y ^ l n 9 } (D) {y |y<0} (E) none of these

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