Questions: Exponential Functions and Graphs

Question 1

The graph of f(x) = 3^x gets closer and closer to the x-axis as x decreases toward negative infinity. What does this tell you about the x-axis?

• A. The x-axis is a vertical asymptote of the function
• B. The x-axis is a horizontal asymptote the graph never actually reaches
• C. The graph eventually crosses the x-axis at a very negative x value
• D. The function becomes zero when x is sufficiently negative

Answer: (B)

For f(x) = 3^x, as x → -∞, f(x) → 0 from above but never equals 0, because 3 raised to any power is always positive. The x-axis (y = 0) is a horizontal asymptote: the graph approaches it indefinitely but never touches or crosses it. This is a fundamental feature of all exponential functions with positive base — the output is always positive.

Question 2

The functions f(x) = (1/2)^x and g(x) = 2^(-x) are different exponential functions with different growth behaviors.

• T. True
• F. False

Answer: False

(1/2)^x = (2^(-1))^x = 2^(-x) by exponent rules. They are the same function. This also reveals a general principle: exponential decay (base between 0 and 1) is the same as growth with a negative exponent. The graph of decay is the reflection of the growth curve across the y-axis.

Question 3

Why does an exponential function like 2^x eventually outgrow any polynomial function like x^100, even though x^100 seems enormous?

Model answer: In a polynomial, the base grows while the exponent is fixed. In an exponential, the exponent grows while the base is fixed. As x increases, the exponent in 2^x keeps increasing, causing the function to repeatedly multiply by 2 — so it compounds indefinitely. A polynomial grows by adding increasingly large terms, but an exponential grows by multiplying, which dominates for large enough x.

The key distinction is what role x plays. In x^100, x is the base — it grows, but each increase only adds one more factor. In 2^x, x is the exponent — each unit increase multiplies the entire previous value by 2. Compounding multiplication always overtakes polynomial growth eventually, no matter how large the polynomial's degree.