## Calculus AB and Calculus BC

## CHAPTER 1 Functions

### Practice Exercises

**Directions:** Answer these questions *without* using your calculator.

**1.** If *f* (*x*) = *x*^{3} − 2*x* − 1, then *f* (−2) =

**(A)** −17

**(B)** −13

**(C)** −5

**(D)** −1

**(E)** 3

**2.** The domain of is

**(A)** all *x* ≠ 1

**(B)** all *x* ≠ 1, −1

**(C)** all *x* ≠ −1

**(D)** *x* 1

**(E)** all reals

**3.** The domain of is

**(A)** all *x* ≠ 0, 1

**(B)** *x* 2, *x* ≠ 0, 1

**(C)** *x* 2

**(D)** *x* 2

**(E)** *x* > 2

**4.** If *f* (*x*) = *x*^{3} − 3*x*^{2} − 2*x* + 5 and *g*(*x*) = 2, then *g*(*f* (*x*)) =

**(A)** 2*x*^{3} − 6*x*^{2} − 2*x* + 10

**(B)** 2*x*^{2} − 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*) = *x*^{3} + *Ax*^{2} + *Bx* − 3 and if *f* (1) = 4 and *f* (−1) = −6, what is the value of 2*A* + *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)**

**(B)** *y* = 2*x*^{4} + 1

**(C)** *y* = *x*^{3} + 2*x*

**(D)** *y* = *x*^{3} + 2

**(E)**

**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* 1}

**(B)** {*y* | − 1 < *y* < 1}

**(C)** {*y*|−1 *y* 1}

**(D)**

**(E)**

**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)**

**(D)** 2

**(E)** π

**11.** Which of the following functions is not odd?

**(A)** *f* (*x*) = sin *x*

**(B)** *f* (*x*) = sin 2*x*

**(C)** *f* (*x*) = *x*^{3} + 1

**(D)**

**(E)**

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

**(A)** 1 and −2

**(B)**

**(C)**

**(D)** 2 and −4

**(E)** −2 and 4

**13.** The set of zeros of *f* (*x*) = *x*^{3} + 4*x*^{2} + 4*x* 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 *y*^{2} = 4*x* 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 − 3* ^{x}* is

**(A)** *g*(*x*) = 1 − 3^{−}^{x}

**(B)** *g*(*x*) = 1 + 3^{x}

**(C)** *g*(*x*) = 3* ^{x}* − 1

**(D)** *g*(*x*) = log_{3} (*x* − 1)

**(E)** *g*(*x*) = log_{3} (1 − *x*)

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

**(A)** 1

**(B)** *x*

**(C)**

**(D)** *f* (*x*) · *g*(*x*)

**(E)** none of these

**17.** The function *f* (*x*) = 2*x*^{3} + *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*) = is

**(A)**

**(B)**

**(C)**

**(D)** 3

**(E)** 6

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

**(A)** {*y* | − ∞ < *y* 0}

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

**(C)** {*y* | −1 < *y* < 1}

**(D)**

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

**20.** If then *b* =

**(A)**

**(B)**

**(C)**

**(D)** 3

**(E)** 9

**21.** Let *f* ^{−1} be the inverse function of *f* (*x*) = *x*^{3} + 2. Then *f* ^{−1}(*x*) =

**(A)**

**(B)** (*x* + 2)^{3}

**(C)** (*x* − 2)^{3}

**(D)**

**(E)**

**22.** The set of *x*-intercepts of the graph of *f* (*x*) = *x*^{3} − 2*x*^{2} − *x* + 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 then the range of *f* (*x*) = *e*^{tan} * ^{x}* 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* = *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 is a maximum is

**(A)**

**(B)** π

**(C)**

**(D)** 3π

**(E)** 6π

**26.**

**(A)** −1

**(B)**

**(C)**

**(D)**

**(E)** 1

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

*f*

^{−1}(

*x*) =

**(A)**

**(B)**

**(C)**

**(D)**

**(E)** ln (2 − *x*)

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

**(A)**

**(B)** *y* = *x*^{3} + 2

**(C)**

**(D)**

**(E)** *y* = ln (*x* − 2) (where *x* >2)

**29.** Suppose that *f* (*x*) = ln *x* for all positive *x* and *g*(*x*) = 9 − *x*^{2} for all real *x*. The domain of *f* (*g*(*x*)) is

**(A)** {*x* | *x* 3}

**(B)** {*x* | |*x*| 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 − *x*^{2} for all real *x*. The range of *y* = *f* (*g*(*x*)) is

**(A)** {*y* | *y* > 0}

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

**(C)** {*y* | *y* ln 9}

**(D)** {*y* | *y* < 0}

**(E)** none of these

**31.** The curve defined parametrically by *x*(*t*) = *t*^{2} + 3 and *y*(*t*) = *t*^{2} + 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*) = cos^{2} (*t*) and *y*(*t*) = 2 sin (*t*)?

**(A)** *x*^{2} + *y*^{2} = 4

**(B)** *x*^{2} + *y*^{2} = 1

**(C)** 4*x*^{2} + *y*^{2} = 4

**(D)** 4*x* + *y*^{2} = 4

**(E)** *x* + 4*y*^{2} = 1

**BC ONLY**

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

**(A)** 0

**(B)**

**(C)**

**(D)**

**(E)**

**BC ONLY**

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

**(A)**

**(B)**

**(C)**

**(D)**

**(E)**

**BC ONLY**

**35.** On the interval [0,2π] there is one point on the curve *r* = − 2 cos 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**