AP Calculus ABPre Calculus ReviewWorksheet # 1

Functions

Concepts and SkillsIn this section we are going to review some basic pre calculus topics as thefollowing:

• General properties of functions: Domain, Range, Composition, Inverse, andso on.

• Special functions and their graphs: absolute value, greatest integer,polynomial, rational, trigonometric, exponential, and logarithmic.

Although these topics are not directly tested on the AP exam, reviewing them willreinforce the calculus comprehension.

80 AP Calculus

Chapter SummaryThis chapter has reviewed some important precalculus topics. These topics are r.directly tested on the AP exam; rather, they represent basic principles important in calcu-lus. These include finding the domain, range and inverse of a function; and understand-ing the properties of polynomial and rational functions, trigonometric and inverse trigfunctions, and exponential and logarithmic functions.

For Be students, this chapter also reviewed parametrically defined functions.

Practice ExercisesDirections: Answer these questions without using your calculator.

1. Iff(x) = x3 - 2x - 1, thenf(-2) =

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

2. The domain off(x) = x2-1 isx + 1

(A) all x:;t 1 (B) "-\allx:;t 1,-1 (C) all x se=-I(D) x~l (E) all reals

3. The domain of g(x) = ..J ~ - 2 isx -x

(A) all x:;t 0, 1 (B) x ~ 2,x:;tO, 1 (C) x~2(D) x~2 (E) x>2

4. If'j'(x) = x3 - 3x2 - 2x + 5 and g(x) = 2, then g(f(x» =

(A) 2x3 - 6x2 - 2x + 10(D) -3 (E) 2

(B) 2X2 - 6x + 1 (C) -6

5. With the functions and choices as in Question 4, which choice is correct forf(g(x»?

6. Iff(x) = x3 +_Ax2+ Bx - 3 and iff(I) = 4 andf(-I) = -6, what is the value of 2A + B?

(A)(E)

12 (B) 8 (C) 0 (D)-2I

It cannot be determined from the given information.

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

x-I(A) y=- (B) y=2~+1 (C) y=x3+2xx

x(D) y=x3+2 (E) y = x3+ 1

8. Let g be a function defined for all reals. Which of the following conditions is notsufficient to guarantee that g has an inverse function?

(A) g(x) = ax + b, a :;t O. {B)(C) g is symmetric to the origin.(E) g is one-to-one.

g is strictly decreasing.(D) g is strictly increasing.

9. Let y = f(x) = sin (arctan x). Then the range offis

Functions 81

(A)

(D)

{YIO<y~l}

{Y 1- ~ <Y< ~}

(B) {YI-l<y<l} (C)

(E) {Y I - ~ ~ Y-~ ~}{YI-l~y~l}

10. Let g(x) = [cos x-II. The maximum value attained by g on the closed interval[0, 2n:] is for x equal to

(A) -1 (B) ° (C) n:2

(D) 2 (E) n:

11. Which of the following functions is not odd?

(A) f(x) = sin x (B) f(x) = sin 2x (C) f(x) =x3 + 1

x(D) f(x) = -2-1

x+(E) f(x) = ifh

12. The roots of the equationf(x) = ° are 1 and -2. The roots off(2x) = ° are

1 1(A) 1 and -2 (B) - and -1 (C) - - and 12 2(D) 2 and-4 (E) -2 and 4

13. The set of zeros off(x) = x3 + 4X2 + 4x is

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

14. The values of x for which the graphs of y = x + 2 and y2 = 4x intersect are

(A) -2 and 2 (B) -2 (C) 2 (D) ° (E) none of theseI

15. The function whose graph is a reflection in the y-axis of the graph off(x) = 113x isI

(A) g(x) = 1 - 3-x (B) g(x) = 1 + 3x (C) g(x) = 3x- 1 I

(D) g(x) = log, (x - 1) (E) g(x) = log, (1 - x)

I16. Letf(x) have an inverse function g(x). Tqenf(g(x)) =

(C) 1x

I

(E) none of these(A) I (B) x

(A) -2 and-1(D) 1 and 2

(B) -landO(E) 2 and 3

I (C) °and 1

17. The functionf(x) = 2x3 + X - 5 has exactly one real zero. It is between

I82 AP Calculus

18. The period off(x) = sin 21t x is3

(A)13

(E) 623

32

(B) (C) (D) 3

19. The range of y =fix) = In (cos x) is

(A) {y1-oo<y~O} (B) {YIO<y~I} (C) {YI-I<y<l}

(D) {YI-~<y<i} (E) {YIO~y~l}

-.

20. bIf log, (3b) = .2' then b =

(A) 1(B)

1 (C) 1 (D) 3 (E) 9- - -9 3 2

21. Letf-i be the inverse function off(x) =.xJ+ 2. Thenf-l(x) =(A) (B) (x + 2)3 (C) (x - 2)3

(D) ~x+2 (E) ~x-2

22. The set of x-intercepts of the graph off(x) =.xJ - 2x2 - X + 2 is

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

23. If the domain off is restricted to the open interval. (-I' I)' then the range off(x) =etan.x is

(A) the set of all reals (B) the set of positive reals(C) the set of nonnegative reals . (D) {y I0 < y ~ 1 J(E) none of these

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

(A) y = -f(x)(D) y = f(lxl)

(B) y = fe-x) (C) y = If(x) I(E) y = -fe-x)

25. The smallest positive x for which the functionf(x) = sin G) - 1 is a maximum is

n 3n(A) 2 (B) n (C) 2 (D) 3n: (E) 6n:

26. tan (arccos (- '?)) =

(A) -1 (B) _ J33

J33

12

(E) 1

29. Suppose thatf(x) = In x for all positive x and g(x) = 9 - X2 for all real x. The domainoff(g(x)) is

(A) {x I x ~ 3}(D) {x Ilxl < 3}

(D)

~ 10. Suppose (as in Question 29) that/ex) = In x for all positive x and g(x) = 9 - X2 for allreal x. The range of y = f(g(x)) is

(C)

(B) circle (C) parabola(E) hyperbola I

I32. Which equation includes the curve defined parametrically by x(t) = cos? (t) and

yet) = 2 sin (t)?

27. Ifr(x) is the inverse off(x) = 2cX, thenf-l(x) =

(A) In (~)

(D) "'In x

(B) in (~)

(E) In (2 -x)

(C) (~)lnx

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

(A) y = sin x (- ~ ~ x ~ ~) (B) y = x3 + 2 (C) y = X2X+ 1

1 xy= -e

2(E) Y = In (x - 2) (where x > 2)(D)

(B) {x Ilxl ~ 3}(E) {x I 0 < x < 3}

(C) {x Ilxl > 3}

(A) {y I y> O}(D) {y l y c O}

(B) {YIO<y~In9}(E) none of these

(C) {y I y ~ In 9}

31. The curve defined parametrically by x(t) = t2 + 3 and yet) = t2 + 4 is part of a(n)

. (A) line(D) ellipse

(A) X2 +i = 4(D) 4x +i = 4

(B) X2 + i = 1 (C) 4X2 + i = 4(E) x + 4y2 = 1

Functions 83