Calculus AB and Calculus BC

CHAPTER 1 Functions

Practice Exercises

Directions: Answer these questions without using your calculator.

1. If f (x) = x3 − 2x − 1, then f (−2) =

(A) −17

(B) −13

(C) −5

(D) −1

(E) 3

2. The domain of Image is

(A) all x ≠ 1

(B) all x ≠ 1, −1

(C) all x ≠ −1

(D) x Image 1

(E) all reals

3. The domain of Image is

(A) all x ≠ 0, 1

(B) x Image 2, x ≠ 0, 1

(C) x Image 2

(D) x Image 2

(E) x > 2

4. If f (x) = x3 − 3x2 − 2x + 5 and g(x) = 2, then g(f (x)) =

(A) 2x3 − 6x2 − 2x + 10

(B) 2x2 − 6 x + 1

(C) −6

(D) −3

(E) 2

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

6. If f (x) = x3 + Ax2 + Bx − 3 and if f (1) = 4 and f (−1) = −6, what is the value of 2A + B ?

(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?

(A)Image

(B) y = 2x4 + 1

(C) y = x3 + 2x

(D) y = x3 + 2

(E) Image

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) g(x) = 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.

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

(A) {y | 0 < y Image 1}

(B) {y | − 1 < y < 1}

(C) {y|−1 Image y Image 1}

(D) Image

(E) Image

10. Let g(x) = |cos x − 1|. The maximum value attained by g on the closed interval [0, 2π] is for x equal to

(A) −1

(B) 0

(C) Image

(D) 2

(E) π

11. Which of the following functions is not odd?

(A) f (x) = sin x

(B) f (x) = sin 2x

(C) f (x) = x3 + 1

(D) Image

(E) Image

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

(A) 1 and −2

(B) Image

(C) Image

(D) 2 and −4

(E) −2 and 4

13. The set of zeros of f (x) = x3 + 4x2 + 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 = 4x 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 f (x) = 1 − 3x is

(A) g(x) = 1 − 3x

(B) g(x) = 1 + 3x

(C) g(x) = 3x − 1

(D) g(x) = log3 (x − 1)

(E) g(x) = log3 (1 − x)

16. Let f (x) have an inverse function g(x). Then f (g(x)) =

(A) 1

(B) x

(C) Image

(D) f (x) · g(x)

(E) none of these

17. The function f (x) = 2x3 + x − 5 has exactly one real zero. It is between

(A) −2 and −1

(B) −1 and 0

(C) 0 and 1

(D) 1 and 2

(E) 2 and 3

18. The period of f (x) = Image is

(A) Image

(B) Image

(C) Image

(D) 3

(E) 6

19. The range of y = f (x) = ln (cos x) is

(A) {y | − ∞ < y Image 0}

(B) {y | 0 < y Image 1}

(C) {y | −1 < y < 1}

(D) Image

(E) {y | 0 Image y Image 1}

20. If Image then b =

(A) Image

(B) Image

(C) Image

(D) 3

(E) 9

21. Let f −1 be the inverse function of f (x) = x3 + 2. Then f −1(x) =

(A) Image

(B) (x + 2)3

(C) (x − 2)3

(D) Image

(E) Image

22. The set of x-intercepts of the graph of f (x) = x3 − 2x2x + 2 is

(A) {1}

(B) {−1,1}

(C) {1,2}

(D) {−1,1,2}

(E) {−1,−2,2}

23. If the domain of f is restricted to the open interval Image then the range of f (x) = etan x is

(A) the set of all reals

(B) the set of positive reals

(C) the set of nonnegative reals

(D) {y | 0 < y Image 1}

(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)

(B) y = f (−x)

(C) y = |f (x)|

(D) y = f (|x|)

(E) y = −f (−x)

25. The smallest positive x for which the function Image is a maximum is

(A) Image

(B) π

(C) Image

(D)

(E)

26. Image

(A) −1

(B) Image

(C) Image

(D) Image

(E) 1

27. If f −1(x) is the inverse of f (x) = 2e−x, then f −1(x) =

(A) Image

(B) Image

(C) Image

(D) Image

(E) ln (2 − x)

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

(A) Image

(B) y = x3 + 2

(C) Image

(D) Image

(E) y = ln (x − 2) (where x >2)

29. Suppose that f (x) = ln x for all positive x and g(x) = 9 − x2 for all real x. The domain of f (g(x)) is

(A) {x | x Image 3}

(B) {x | |x| Image 3}

(C) {x | |x| > 3}

(D) {x | |x| < 3}

(E) {x | 0 < x < 3}

30. Suppose (as in Question 29) that f (x) = ln x for all positive x and g(x) = 9 − x2 for all real x. The range of y = f (g(x)) is

(A) {y | y > 0}

(B) {y | 0 < y Image ln 9}

(C) {y | y Image ln 9}

(D) {y | y < 0}

(E) none of these

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

(A) line

(B) circle

(C) parabola

(D) ellipse

(E) hyperbola

BC ONLY

32. Which equation includes the curve defined parametrically by x(t) = cos2 (t) and y(t) = 2 sin (t)?

(A) x2 + y2 = 4

(B) x2 + y2 = 1

(C) 4x2 + y2 = 4

(D) 4x + y2 = 4

(E) x + 4y2 = 1

BC ONLY

33. Find the smallest value of Image in the interval [0,2π] for which the rose r = 2 cos(5Image) passes through the origin.

(A) 0

(B) Image

(C) Image

(D) Image

(E) Image

BC ONLY

34. For what value of Image in the interval [0,π] do the polar curves r = 3 and r = 2 + 2 cos Image intersect?

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

BC ONLY

35. On the interval [0,2π] there is one point on the curve r = Image − 2 cos Image whose x-coordinate is 2. Find the y-coordinate there.

(A) −4.594

(B) −3.764

(C) 1.979

(D) 4.263

(E) 5.201

BC ONLY