PRE CALCULUS Exponential and Logarithmic Functions 1 — Questions and Answers
Question 1: Which of the following is the general form of an exponential function?
- f(x) = x^b
- f(x) = a·b^x where b > 0, b ≠ 1 (Correct answer)
- f(x) = a·x^b where a ≠ 0
- f(x) = log_b(x)
Correct answer: f(x) = a·b^x where b > 0, b ≠ 1
An exponential function has the form f(x) = a·b^x where b is a positive base not equal to 1 and x is the exponent.
Question 2: What is the y-intercept of the function f(x) = 3·2^x?
- 0
- 2
- 3 (Correct answer)
- 6
Correct answer: 3
At the y-intercept, x = 0, so f(0) = 3·2^0 = 3·1 = 3.
Question 3: Which statement describes the end behavior of f(x) = (1/2)^x as x → ∞?
- f(x) → ∞
- f(x) → 0 (Correct answer)
- f(x) → 1
- f(x) → −∞
Correct answer: f(x) → 0
Since the base 1/2 is between 0 and 1, repeated multiplication makes (1/2)^x approach 0 as x increases.
Question 4: What is the value of e^0?
- 0
- e
- 1 (Correct answer)
- undefined
Correct answer: 1
Any nonzero number raised to the power of 0 equals 1, so e^0 = 1.
Question 5: The exponential function f(x) = 5^x is a one-to-one function. What does this imply?
- It has no inverse
- Its inverse is also a function (Correct answer)
- It passes the horizontal line test only sometimes
- Its range is all real numbers
Correct answer: Its inverse is also a function
Because f(x) = 5^x is one-to-one (passes the horizontal line test), its inverse (the logarithm) is also a valid function.
Question 6: If f(x) = 2^x, what is f(x + 3) in simplified form?
- 2^x + 8
- 8·2^x (Correct answer)
- 2^x + 3
- 6·2^x
Correct answer: 8·2^x
f(x + 3) = 2^(x+3) = 2^x · 2^3 = 8·2^x using the product rule of exponents.
Question 7: Which of the following represents exponential growth rather than decay?
- f(x) = 4·(0.5)^x
- f(x) = 2·(0.9)^x
- f(x) = 3·(1.2)^x (Correct answer)
- f(x) = 5·(0.1)^x
Correct answer: f(x) = 3·(1.2)^x
Exponential growth occurs when the base b > 1; in f(x) = 3·(1.2)^x, the base 1.2 is greater than 1.
PRE CALCULUS Exponential and Logarithmic Functions 1
Which of the following is the general form of an exponential function?