Rucete ✏ AP Precalculus In a Nutshell
5. Exponential Functions — Practice Questions 3
This chapter covers the properties, graphs, transformations, and applications of exponential functions, including continuous growth and decay, half-life, regression models, and real-world contexts.
(Multiple Choice — Click to Reveal Answer)
1. Which of the following is an exponential function?
(A) f(x) = 2x + 3
(B) f(x) = x^2
(C) f(x) = 3^x
(D) f(x) = log(x)
Answer
(C) — The variable appears in the exponent, which defines an exponential function.
2. The graph of f(x) = 2^x has which asymptote?
(A) x = 0
(B) y = 0
(C) y = 2
(D) x = 2
Answer
(B) — Exponential functions always have a horizontal asymptote at y = 0 unless shifted vertically.
3. Which statement is true about exponential decay?
(A) The base b > 1
(B) The base 0 < b < 1
(C) The exponent is always negative
(D) The graph passes through (0, 0)
Answer
(B) — For decay, the base must be between 0 and 1.
4. The function f(x) = 3^x has a y-intercept at:
(A) (0, 1)
(B) (0, 0)
(C) (1, 3)
(D) (3, 0)
Answer
(A) — At x=0, f(0)=1.
5. Which of the following best describes the domain of exponential functions f(x)=a·b^x (a≠0, b>0, b≠1)?
(A) x ≥ 0
(B) All real numbers
(C) x ≠ 0
(D) x > 0
Answer
(B) — The exponent can take any real value.
6. Which property explains why 2^(x+3) = 8·2^x?
(A) Power of a Power
(B) Negative Exponent Law
(C) Product Property
(D) Quotient Property
Answer
(C) — b^(m+n) = b^m · b^n.
7. Which transformation occurs when comparing f(x)=2^x and g(x)=2^(x−4)?
(A) Shift right 4 units
(B) Shift left 4 units
(C) Stretch vertically by factor 4
(D) Reflect over y-axis
Answer
(A) — Subtracting inside the exponent shifts the graph to the right.
8. The function f(x)=5^(−x) is a reflection of f(x)=5^x over:
(A) x-axis
(B) y-axis
(C) line y=x
(D) origin
Answer
(B) — Negative exponent reflects across the y-axis.
9. Which is equivalent to (3^x)^2?
(A) 3^(2x)
(B) 6^x
(C) 9^x
(D) 3^(x^2)
Answer
(A) — (a^x)^y = a^(xy).
10. The continuous growth formula is f(t)=ae^(kt). What does k represent?
(A) The initial amount
(B) The time variable
(C) The growth or decay rate constant
(D) The base of natural logs
Answer
(C) — k is the continuous growth or decay rate.
11. The range of f(x)=4^x is:
(A) (−∞, ∞)
(B) (0, ∞)
(C) [0, ∞)
(D) (−∞, 0)
Answer
(B) — Exponential functions always produce positive values.
12. Which point lies on the graph of f(x)=10^x?
(A) (1, 10)
(B) (0, 10)
(C) (−1, 10)
(D) (10, 1)
Answer
(A) — f(1)=10^1=10.
13. What is the end behavior of f(x)=(1/3)^x as x→∞?
(A) f(x)→∞
(B) f(x)→0
(C) f(x) oscillates
(D) f(x)→−∞
Answer
(B) — With base between 0 and 1, the function decays toward 0.
14. Which is the inverse of f(x)=2^x?
(A) f(x)=log_2(x)
(B) f(x)=ln(x)
(C) f(x)=1/(2^x)
(D) f(x)=x^2
Answer
(A) — The inverse of an exponential is a logarithm with the same base.
15. The graph of f(x)=e^x passes through which point?
(A) (1, e)
(B) (0, e)
(C) (e, 1)
(D) (1, 0)
Answer
(A) — At x=1, f(1)=e.
16. If f(x)=2^x, what is f(−2)?
(A) 1/2
(B) 1/4
(C) 2
(D) 4
Answer
(B) — f(−2)=2^(−2)=1/4.
17. Which transformation occurs when f(x)=3^x changes to g(x)=−3^x?
(A) Vertical shift
(B) Reflection across x-axis
(C) Reflection across y-axis
(D) Horizontal shift
Answer
(B) — Negative outside reflects across the x-axis.
18. Which exponential function represents decay?
(A) f(x)=5^x
(B) f(x)=(1/5)^x
(C) f(x)=7^x
(D) f(x)=10^x
Answer
(B) — Base between 0 and 1 indicates decay.
19. If a population doubles every 5 years, which model fits best?
(A) P(t)=P0·2^(t/5)
(B) P(t)=P0·(1/2)^t
(C) P(t)=P0·5^t
(D) P(t)=P0·2t
Answer
(A) — The doubling time formula uses base 2 with exponent (t/period).
20. What is f(0) if f(x)=7^x?
(A) 0
(B) 1
(C) 7
(D) −1
Answer
(B) — Any nonzero base to power 0 equals 1.
21. Which function is equivalent to 4^(x+1)?
(A) 4^x+1
(B) 5·4^x
(C) 4·4^x
(D) 4^(x)−1
Answer
(C) — 4^(x+1)=4·4^x.
22. The half-life formula uses which base?
(A) 2
(B) 1/2
(C) e
(D) ln
Answer
(B) — Half-life uses (1/2)^(t/period).
23. Which is NOT an exponential function?
(A) y=5^x
(B) y=(1/3)^x
(C) y=x^5
(D) y=e^x
Answer
(C) — The variable must be in the exponent, not the base.
24. If f(x)=10^x, what is log_10(f(x))?
(A) 1
(B) x
(C) 10
(D) e
Answer
(B) — log and exponential with same base are inverses.
25. Which describes the growth rate of f(x)=1.1^x?
(A) 1%
(B) 10%
(C) 11%
(D) 0.1%
Answer
(B) — The base is 1+r, so r=0.1=10%.
26. Solve for x: 2^(x+1)=16.
(A) 2
(B) 3
(C) 4
(D) 5
Answer
(B) — 2^(x+1)=16 → 2^(x+1)=2^4 → x+1=4 → x=3.
27. Which function represents exponential decay with initial value 200 and half-life of 10 years?
(A) f(t)=200·(1/2)^(t/10)
(B) f(t)=200·(2)^t
(C) f(t)=200·(1/10)^t
(D) f(t)=200·(1/2)^t
Answer
(A) — Half-life model uses (1/2)^(t/period).
28. If f(x)=e^x and g(x)=e^(2x), what is g(x)/f(x)?
(A) e^x
(B) e^(x/2)
(C) e^(2x−x)=e^x
(D) e^(3x)
Answer
(A)/(C) — g(x)/f(x)=e^(2x)/e^x=e^x.
29. Which equation is equivalent to ln(5)=x?
(A) e^5=x
(B) e^x=5
(C) log_5(x)=e
(D) 5^x=e
Answer
(B) — ln(5)=x means e^x=5.
30. Solve for x: 3^(2x)=81.
(A) 2
(B) 3
(C) 4
(D) 5
Answer
(A) — 3^(2x)=81=3^4 → 2x=4 → x=2.
31. Which is the horizontal asymptote of f(x)=5^x+7?
(A) y=5
(B) y=7
(C) y=0
(D) y=12
Answer
(B) — Base exponential has asymptote y=0, vertical shift +7 → asymptote y=7.
32. Which expression equals log(1000)?
(A) 2
(B) 3
(C) 10
(D) 100
Answer
(B) — log base 10, log(1000)=3.
33. If ln(x)=2, what is x?
(A) e
(B) e^2
(C) 2e
(D) 4
Answer
(B) — ln(x)=2 → x=e^2.
34. Which function grows faster as x→∞?
(A) f(x)=2^x
(B) g(x)=3^x
(C) h(x)=x^10
(D) k(x)=ln(x)
Answer
(B) — Higher base exponential dominates growth rate.
35. Solve for x: e^(x−1)=20.
(A) ln(20)+1
(B) ln(20)−1
(C) ln(21)
(D) 20e
Answer
(A) — x−1=ln(20) → x=ln(20)+1.
36. Solve for x: 5^x = 125
Answer
x=3, since 125=5^3.
37. Find the domain of f(x)=ln(x−4).
Answer
x>4, because argument of ln must be positive.
38. Evaluate log_2(32).
Answer
5, since 2^5=32.
39. Solve for x: e^(2x)=7.
Answer
x=(1/2)·ln(7).
40. If f(x)=2^x and g(x)=3^x, find f(2)+g(2).
Answer
2^2+3^2=4+9=13.
41. Solve for t: 1000·(1.05)^t=2000.
Answer
(1.05)^t=2 → t=ln(2)/ln(1.05)≈14.2.
42. What is the y-intercept of f(x)=e^(x+2)?
Answer
(0,e^2).
43. Solve for x: ln(x)=5.
Answer
x=e^5.
44. Simplify: log(100) + log(1000).
Answer
2+3=5.
45. Solve for x: 4^(x+1)=64.
Answer
4^(x+1)=4^3 → x+1=3 → x=2.
46. If a population grows by 8% annually, write its exponential model P(t).
Answer
P(t)=P0·(1.08)^t.
47. Find the range of f(x)=e^(−x).
Answer
(0,∞).
48. Solve for x: log_5(x)=4.
Answer
x=5^4=625.
49. If f(x)=2^x, evaluate f(−3).
Answer
2^(−3)=1/8.
50. A radioactive substance has half-life 12 hours. If starting with 200 g, how much remains after 36 hours?
Answer
200·(1/2)^(36/12)=200·(1/2)^3=200·(1/8)=25 g.