Rucete ✏ AP Precalculus In a Nutshell
5. Exponential Functions — Practice Questions
This chapter introduces the properties, graphs, and applications of exponential functions, including growth, decay, transformations, and real-world modeling.
(Multiple Choice — Click to Reveal Answer)
1. Which of the following is the general form of an exponential function?
A) f(x) = ax + b
B) f(x) = ab^x
C) f(x) = ax^b
D) f(x) = log_a(x)
Answer
B) f(x) = ab^x, where a ≠ 0 and b > 0, b ≠ 1 :contentReference[oaicite:0]{index=0}
2. What is the domain of f(x) = 2^x?
A) (0, ∞)
B) (−∞, ∞)
C) [0, ∞)
D) (−∞, 0)
Answer
B) (−∞, ∞). Exponential functions are defined for all real x :contentReference[oaicite:1]{index=1}
3. Which of the following describes the range of exponential functions with a > 0?
A) (−∞, ∞)
B) (0, ∞)
C) (−∞, 0)
D) [0, ∞)
Answer
B) (0, ∞). Exponential functions never take negative values :contentReference[oaicite:2]{index=2}
4. The function f(x) = 2^x has which y-intercept?
A) (0, 0)
B) (0, 1)
C) (1, 2)
D) (0, 2)
Answer
B) (0, 1). Substituting x = 0 gives f(0) = 2^0 = 1 :contentReference[oaicite:3]{index=3}
5. Which of the following describes the end behavior of f(x) = 3^x?
A) As x → ∞, f(x) → 0; as x → −∞, f(x) → ∞
B) As x → ∞, f(x) → ∞; as x → −∞, f(x) → 0
C) As x → ∞, f(x) → ∞; as x → −∞, f(x) → −∞
D) As x → ∞, f(x) → 0; as x → −∞, f(x) → −∞
Answer
B) As x → ∞, f(x) → ∞; as x → −∞, f(x) → 0 :contentReference[oaicite:4]{index=4}
6. Which of the following functions represents exponential decay?
A) f(x) = 5(2^x)
B) f(x) = 3(0.5^x)
C) f(x) = 4x^2
D) f(x) = log(x)
Answer
B) f(x) = 3(0.5^x). Since 0 < base < 1, this is exponential decay.
7. What is the horizontal asymptote of f(x) = 2^x + 4?
A) y = 0
B) y = 2
C) y = 4
D) y = −4
Answer
C) y = 4. The graph of 2^x is shifted upward by 4.
8. The graph of f(x) = −3(2^x) is…
A) Exponential growth, reflected across the y-axis
B) Exponential decay, reflected across the x-axis
C) Exponential growth, reflected across the x-axis
D) Exponential decay, reflected across the y-axis
Answer
C) Exponential growth, reflected across the x-axis due to the negative coefficient.
9. Which of the following passes through the point (0, 5)?
A) f(x) = 5^x
B) f(x) = 5(2^x)
C) f(x) = 2^x + 5
D) f(x) = 5(0.5^x)
Answer
B) f(x) = 5(2^x). At x=0, f(0)=5(1)=5.
10. Which of the following best describes the function f(x) = e^x?
A) Linear with slope e
B) Exponential growth with base e
C) Logarithmic with base e
D) Quadratic with vertex (0,1)
Answer
B) Exponential growth with base e ≈ 2.718.
11. Which has a faster growth rate as x → ∞?
A) 2^x
B) 3^x
C) e^x
D) x^5
Answer
B) 3^x grows faster than both 2^x and polynomials like x^5.
12. If f(x) = 4^x, then f(−2) equals…
A) 16
B) 1/16
C) −16
D) 1/4
Answer
B) f(−2) = 1/4^2 = 1/16.
13. Which function has y-intercept (0, 7)?
A) f(x) = 7^x
B) f(x) = 2^x + 7
C) f(x) = 7(3^x)
D) f(x) = log(x)+7
Answer
C) f(x) = 7(3^x). At x=0, f(0)=7(1)=7.
14. The half-life of a substance can be modeled with…
A) Linear functions
B) Exponential decay functions
C) Quadratic functions
D) Trigonometric functions
Answer
B) Exponential decay functions model half-life.
15. Which of the following shows exponential growth with initial value 2 and growth factor 3?
A) f(x) = 2^3x
B) f(x) = 2(3^x)
C) f(x) = 3(2^x)
D) f(x) = (2^x)(3^x)
Answer
B) f(x) = 2(3^x). Initial value 2, base 3 growth.
16. For f(x) = (1/3)^x, which statement is true?
A) Exponential growth, y→0 as x→−∞
B) Exponential decay, y→0 as x→∞
C) Linear decay
D) No asymptote
Answer
B) Exponential decay; as x→∞, y→0.
17. Which of the following represents a horizontal shift of f(x) = 2^x to the right by 3?
A) f(x) = 2^(x−3)
B) f(x) = 2^(x+3)
C) f(x) = (2^x)+3
D) f(x) = 3(2^x)
Answer
A) f(x) = 2^(x−3). Subtracting inside shifts right.
18. Which function decays faster?
A) (1/2)^x
B) (3/4)^x
C) (4/5)^x
D) (9/10)^x
Answer
A) (1/2)^x decays fastest since the base is smallest.
19. If f(x) = 2^x and g(x) = 2^(x+1), then g(x) = ?
A) f(x)+1
B) 2f(x)
C) f(x)+2
D) f(x−1)
Answer
B) g(x) = 2^(x+1) = 2·2^x = 2f(x).
20. Which of the following represents compound interest?
A) A = P(1+rt)
B) A = P(1+r)^t
C) A = P + rt
D) A = Pe^(rt)
Answer
D) A = Pe^(rt) is continuous compound interest; B is discrete compounding. Both model exponential growth.
21. What is the base of the natural exponential function?
A) 2
B) 10
C) e
D) 0
Answer
C) The base is e ≈ 2.718.
22. Which transformation occurs in f(x) = 2^x − 5?
A) Shift left 5
B) Shift right 5
C) Shift down 5
D) Shift up 5
Answer
C) Subtracting outside shifts the graph down 5.
23. Which has a vertical stretch by factor 4?
A) f(x) = 4(2^x)
B) f(x) = 2^(4x)
C) f(x) = (2^x)+4
D) f(x) = 2^(x+4)
Answer
A) Multiplying outside by 4 is a vertical stretch.
24. Which of the following is true for all exponential functions f(x) = ab^x (a>0, b>0, b≠1)?
A) They pass through (0,a)
B) They pass through (1,b)
C) They pass through (0,1)
D) They are symmetric about y=x
Answer
A) At x=0, f(0)=a·b^0=a.
25. If f(x) = 2^x and g(x) = 3^x, then which grows faster as x→∞?
A) f(x)
B) g(x)
C) Both grow equally
D) Neither grows
Answer
B) g(x)=3^x grows faster than f(x)=2^x.
26. The population of a bacteria culture doubles every 6 hours. If the initial population is 500, which function models the population after t hours?
A) P(t) = 500(2^t)
B) P(t) = 500(2^(t/6))
C) P(t) = 500(2^(6t))
D) P(t) = 500(0.5^(t/6))
Answer
B) Since it doubles every 6 hours, growth factor is 2^(t/6).
27. The half-life of a radioactive substance is 10 years. If the initial amount is 200 g, which equation models the amount after t years?
A) A(t) = 200(0.5^t)
B) A(t) = 200(0.5^(t/10))
C) A(t) = 200(2^(t/10))
D) A(t) = 200e^(10t)
Answer
B) Half-life means the base is 0.5 with exponent t/10.
28. Solve for t: 500e^(0.03t) = 1000.
A) t = ln(2)/0.03
B) t = ln(1000)/0.03
C) t = ln(500)/0.03
D) t = 2/0.03
Answer
A) Divide both sides: e^(0.03t) = 2 ⇒ 0.03t = ln(2).
29. Which function has the same y-intercept as f(x) = 3^x but grows faster?
A) g(x) = 2^x
B) g(x) = 5^x
C) g(x) = e^x
D) g(x) = 3^(x+1)
Answer
B) Both f and g have y-intercept (0,1), but 5^x grows faster.
30. For f(x) = e^(−0.2x), what is the horizontal asymptote?
A) y = 0
B) y = −0.2
C) y = e
D) y = 1
Answer
A) All exponential functions of the form e^(−kx) approach 0 as x→∞.
31. The function f(x) = 100(1.05)^x represents a savings account. What is the annual percentage growth rate?
A) 0.05%
B) 5%
C) 50%
D) 500%
Answer
B) Growth factor is 1.05 ⇒ growth rate 5%.
32. Which equation represents exponential decay with a rate of 12% per year and initial value 800?
A) A(t) = 800(0.12^t)
B) A(t) = 800(0.88^t)
C) A(t) = 800(1.12^t)
D) A(t) = 800e^(0.12t)
Answer
B) Since decay rate is 12%, multiply by (1−0.12)=0.88 each year.
33. Which of the following has the steepest initial growth?
A) y = 2^x
B) y = 5^x
C) y = e^x
D) y = 10^x
Answer
D) Higher base grows steepest; 10^x grows faster than 5^x and 2^x.
34. Solve for x: 4^(x+1) = 64.
A) x = 1
B) x = 2
C) x = 3
D) x = 4
Answer
B) 4^(x+1)=64 → 4^3=64 → x+1=3 → x=2.
35. Which best describes the function f(x) = −2(3^x) + 5?
A) Growth, shifted up 5
B) Decay, shifted up 5
C) Growth, reflected over x-axis, shifted up 5
D) Growth, reflected over y-axis, shifted right 5
Answer
C) Negative sign reflects across x-axis; +5 shifts up.
36. Find the value of x if 2^x = 128.
Answer
2^x = 128 ⇒ 128 = 2^7 ⇒ x = 7.
37. Solve for x: e^(2x) = 20.
Answer
Take ln: 2x = ln(20) ⇒ x = ln(20)/2.
38. The population of a town is modeled by P(t) = 5000(1.04)^t. Find the population after 5 years.
Answer
P(5) = 5000(1.04)^5 ≈ 5000(1.21665) ≈ 6083.
39. If f(x) = 3^x, find f(−3).
Answer
f(−3) = 3^(−3) = 1/27.
40. Solve for t: 200(0.8^t) = 50.
Answer
0.8^t = 50/200 = 0.25 ⇒ t = log(0.25)/log(0.8) ≈ 6.21.
41. A radioactive sample decays according to A(t) = 100e^(−0.05t). Find A(20).
Answer
A(20) = 100e^(−1) ≈ 100(0.3679) ≈ 36.8.
42. Determine the horizontal asymptote of f(x) = 7(0.9^x) − 2.
Answer
As x→∞, 0.9^x→0 ⇒ f(x)→ −2. Horizontal asymptote y = −2.
43. If g(x) = 10(2^x), find g(4).
Answer
g(4) = 10(2^4) = 10(16) = 160.
44. Solve: 5^x = 1/125.
Answer
1/125 = 5^(−3). Thus, x = −3.
45. A culture of bacteria triples every 12 hours. If initially 200, how many after 24 hours?
Answer
After 24 hours = 200(3^(24/12)) = 200(3^2) = 200(9) = 1800.
46. Solve for x: ln(x) = 3.
Answer
x = e^3 ≈ 20.085.
47. If f(x) = e^(−x), find f(ln(5)).
Answer
f(ln(5)) = e^(−ln(5)) = 1/5.
48. The value of a car decreases by 15% each year. If the car is worth $20,000 now, what is its value after 3 years?
Answer
Value = 20000(0.85^3) = 20000(0.614125) ≈ 12,282.5.
49. Find the doubling time for a population modeled by P(t) = 1000e^(0.07t).
Answer
2 = e^(0.07t) ⇒ t = ln(2)/0.07 ≈ 9.90.
50. Solve for x: log_2(x) = 5.
Answer
x = 2^5 = 32.