Rucete ✏ AP Precalculus In a Nutshell
5. Exponential Functions — Practice Questions 2
This chapter explores the properties, transformations, and applications of exponential functions, including growth, decay, half-life, and continuous compounding.
(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) = x^a
(D) f(x) = log_b(x)
Answer
(B) — Exponential functions have a constant base b with the variable in the exponent.
2. The domain of f(x) = 2^x is:
(A) x > 0
(B) x ≥ 0
(C) all real numbers
(D) x ≠ 0
Answer
(C) — Exponential functions are defined for all real x.
3. The range of f(x) = 3^x is:
(A) (0, ∞)
(B) (−∞, ∞)
(C) [0, ∞)
(D) (−∞, 0)
Answer
(A) — Outputs are always positive for exponential functions.
4. What is the y-intercept of f(x) = 5^x ?
(A) (0,0)
(B) (0,1)
(C) (0,5)
(D) (1,5)
Answer
(B) — At x=0, f(0)=1.
5. Which describes f(x) = (1/2)^x ?
(A) Growth with base >1
(B) Decay approaching 0
(C) Linear decrease
(D) Oscillating function
Answer
(B) — Since 0<base<1, the function decays toward 0 as x increases.
6. Which of the following is always the horizontal asymptote of an exponential function f(x) = ab^x?
(A) x = 0
(B) y = 0
(C) y = a
(D) x = a
Answer
(B) — Exponential functions always approach y=0 as x → −∞ if a>0.
7. For f(x) = 4^x, what is f(−2)?
(A) 16
(B) 1/16
(C) 1/4
(D) 1/2
Answer
(B) — f(−2) = 4^(−2) = 1/16.
8. Which graph shows exponential decay?
(A) f(x) = 3^x
(B) f(x) = 5^x
(C) f(x) = (1/2)^x
(D) f(x) = 7^x
Answer
(C) — Bases between 0 and 1 represent decay.
9. Which of the following is NOT true for f(x)=2^x?
(A) Domain is all real numbers
(B) Range is (0,∞)
(C) Graph passes through (0,1)
(D) Graph is symmetric about the y-axis
Answer
(D) — 2^x is not symmetric; it's increasing, not even or odd.
10. If g(x)=2^x+3, what is its horizontal asymptote?
(A) y=0
(B) y=1
(C) y=3
(D) y=−3
Answer
(C) — Vertical shift of +3 moves asymptote up to y=3.
11. Which equation models continuous growth at 8%?
(A) f(t)=P(1.08)^t
(B) f(t)=Pe^(0.08t)
(C) f(t)=P(0.92)^t
(D) Both A and B
Answer
(D) — Discrete growth uses (1.08)^t, continuous uses e^(0.08t).
12. A bacteria culture starts at 200 and doubles every 6 hours. Which model is correct?
(A) P(t)=200·2^t
(B) P(t)=200·2^(t/6)
(C) P(t)=200·6^t
(D) P(t)=200·(1/2)^t
Answer
(B) — Doubling every 6 hours means exponent is t/6.
13. The inverse of f(x)=10^x is:
(A) f^−1(x)=log_10(x)
(B) f^−1(x)=ln(x)
(C) f^−1(x)=x^10
(D) f^−1(x)=10^(1/x)
Answer
(A) — The inverse of 10^x is log base 10.
14. Solve for x: 2^(x+1)=32.
(A) 3
(B) 4
(C) 5
(D) 6
Answer
(B) — 32=2^5 → x+1=5 → x=4.
15. Which of the following describes f(x)=(1/3)^x?
(A) Growth, concave up
(B) Growth, concave down
(C) Decay, concave up
(D) Decay, concave down
Answer
(C) — Bases 0<b<1 give decay; exponential graphs are concave up.
16. Which transformation shifts f(x)=3^x to the left by 2 units?
(A) 3^(x−2)
(B) 3^(x+2)
(C) 3^x+2
(D) −3^x
Answer
(B) — Adding inside (x+2) shifts the graph left by 2.
17. Which exponential function has y-intercept 4?
(A) 4^x
(B) 4·(1.5^x)
(C) (1.5^x)+4
(D) (4^x)+1
Answer
(B) — For f(x)=a·b^x, f(0)=a. Here a=4 → y-intercept (0,4).
18. Evaluate f(−3) for f(x)=(1/2)^x.
(A) 1/8
(B) 1/6
(C) 8
(D) 6
Answer
(C) — (1/2)^(−3)=2^3=8.
19. Solve for t: 8·(1/2)^t = 1.
(A) t=1
(B) t=2
(C) t=3
(D) t=4
Answer
(C) — (1/2)^t=1/8=(1/2)^3 → t=3.
20. Which identity is correct for all real x?
(A) e^x · e^{2x} = e^{x^2}
(B) e^x · e^{2x} = e^{3x}
(C) e^x + e^{2x} = e^{3x}
(D) e^{x+2x} = e^{x^2}
Answer
(B) — Same base multiply → add exponents.
21. For f(x)=a·b^x with a>0, b>0, b≠1, which point is always on the graph?
(A) (0,0)
(B) (1,0)
(C) (0,a)
(D) (a,0)
Answer
(C) — f(0)=a·b^0=a → passes through (0,a).
22. Let g(x)=2^x+2 and h(x)=2^{x+1}+2. Which relation is true?
(A) h(x)=g(x)+1
(B) h(x)=2g(x)
(C) h(x)=2g(x)−2
(D) h(x)=g(x)/2
Answer
(C) — h=2·2^x+2 and 2g−2=2·2^x+4−2=2·2^x+2.
23. For f(x)=0.8^x, which describes the end behavior?
(A) x→∞ ⇒ f(x)→∞
(B) x→∞ ⇒ f(x)→0
(C) x→−∞ ⇒ f(x)→0
(D) Both B and C
Answer
(B) — With 0<b<1, as x increases, f(x) decays to 0. (As x→−∞, f(x)→∞.)
24. Find the exponential model f(x)=a·b^x passing through (0,3) and (1,9).
(A) 3·(2^x)
(B) 3·(3^x)
(C) 9·(3^x)
(D) 3^x+3
Answer
(B) — From (0,3): a=3. From (1,9): 3·b=9 → b=3 → f(x)=3·3^x.
25. Simplify: 7^x · 7^{−2} = ?
(A) 7^{x+2}
(B) 7^{x−2}
(C) 7^{2−x}
(D) 7^{−x−2}
Answer
(B) — Same base multiply → add exponents: x + (−2) = x−2.
26. Which of the following functions has both a hole and a vertical asymptote?
(A) (x−2)(x+3)/(x−2)(x−4)
(B) (x^2+1)/(x+5)
(C) (x−1)/(x^2+1)
(D) (x^2−4)/(x^2+9)
Answer
(A) — Factor (x−2) cancels → hole at x=2; denominator still has (x−4) → vertical asymptote at x=4.
27. Find the slant asymptote of f(x) = (2x^3+5x^2−1)/(x^2+1).
(A) y = 2x+5
(B) y = 2x+3
(C) y = 2x+1
(D) y = x+2
Answer
(A) — Polynomial long division gives quotient 2x+5.
28. For g(x) = (x^2−4)/(x^2−9), what is the horizontal asymptote?
(A) y=0
(B) y=1
(C) y=2
(D) none
Answer
(B) — Degrees equal; ratio of leading coefficients 1/1=1.
29. Which rational function has oblique asymptote y=3x−2?
(A) (3x^2−2x+1)/(x+1)
(B) (3x^2−2)/(x+2)
(C) (3x^2+1)/(x−1)
(D) (3x^2−6x+2)/(x+1)
Answer
(A) — Division yields quotient 3x−2 with remainder.
30. For h(x) = (x^3−2x)/(x^2−1), which are vertical asymptotes?
(A) x=±1
(B) x=0, x=1
(C) x=2
(D) none
Answer
(A) — Denominator factors (x−1)(x+1) → vertical asymptotes at ±1.
31. Which function’s graph intersects its slant asymptote?
(A) (x^2+1)/(x+1)
(B) (3x^3+1)/(x^2+2)
(C) (2x^2+1)/(x+2)
(D) (x^3+2)/(x^2+1)
Answer
(D) — Higher numerator degree leads to crossing behavior.
32. The domain of f(x) = (x^2−9)/(x^2−x−6) is:
(A) all real x
(B) x≠3 only
(C) x≠−2,3
(D) x≠±3
Answer
(C) — Factor denominator: (x−3)(x+2). Domain excludes x=3,−2.
33. As x→∞, f(x)=(5x^3−x)/(2x^3+7) approaches:
(A) 5/2
(B) ∞
(C) 0
(D) 7/2
Answer
(A) — Leading coefficient ratio 5/2.
34. Which describes end behavior of f(x) = (x^4+2)/(x^2+1)?
(A) Horizontal asymptote y=1
(B) Slant asymptote y=x^2
(C) Quadratic asymptote y=x^2
(D) none
Answer
(C) — Degree difference 2; asymptote is quadratic y=x^2.
35. Which rational inequality solution is correct? (x−1)/(x+2) > 0
(A) (−∞,−2)∪(1,∞)
(B) (−2,1)
(C) (−∞,−2)∪(−2,1)
(D) (1,∞)
Answer
(A) — Critical points −2 (VA) and 1 (zero). Sign analysis shows positivity on (−∞,−2) and (1,∞).
36. Find the domain of f(x) = 1/(x^2−4).
Answer
Exclude values that make denominator 0. x≠±2 → Domain: (−∞,−2)∪(−2,2)∪(2,∞).
37. Determine the range of f(x)=2^x−3.
Answer
Exponential range is (0,∞). Shifted down by 3 → Range: (−3,∞).
38. Find the x-intercept of f(x)=(x−1)/(x+2).
Answer
Set numerator=0 → x=1. Intercept at (1,0).
39. Write the inverse of f(x)=3^x.
Answer
Inverse is f^−1(x)=log_3(x).
40. Find vertical asymptotes of f(x)=(2x+1)/(x^2−9).
Answer
Denominator zeros: x=±3. Vertical asymptotes at x=−3,3.
41. Solve for x: 5^(x−1)=25.
Answer
25=5^2 → x−1=2 → x=3.
42. Find y-intercept of f(x)=(x^2−1)/(x^2+1).
Answer
At x=0: (−1)/(1)=−1. y-intercept (0,−1).
43. Determine if f(x)=x^3/(x^2+1) is odd, even, or neither.
Answer
f(−x)=(−x)^3/((−x)^2+1)=−x^3/(x^2+1)=−f(x). → Odd.
44. Evaluate the limit: lim (x→∞) (5x^2+1)/(2x^2−3).
Answer
Leading coefficients ratio: 5/2. Limit=5/2.
45. Find oblique asymptote of f(x)=(x^2+3)/(x−1).
Answer
Divide: (x^2+3)/(x−1)=x+1+4/(x−1). Slant asymptote y=x+1.
46. Solve inequality: (x+1)/(x−2) < 0.
Answer
Critical points: −1 (zero), 2 (VA). Test intervals: solution (−1,2).
47. Find the hole of f(x)=(x^2−1)/(x^2−x−2).
Answer
Factor numerator (x−1)(x+1), denominator (x−2)(x+1). (x+1) cancels → hole at x=−1. Evaluate limit → (−1−1)/(−1−2)=(−2)/(−3)=2/3. Hole at (−1,2/3).
48. Find range of f(x)=e^x+5.
Answer
e^x has range (0,∞). Shift up 5 → Range (5,∞).
49. Solve for x: log_2(x)=5.
Answer
x=2^5=32.
50. A radioactive substance decays with half-life 10 years. Initial 80g. Find amount after 30 years.
Answer
Formula: A(t)=A0·(1/2)^(t/half-life). A(30)=80·(1/2)^(30/10)=80·(1/8)=10 g.