Question

Which exponential function has an initial value of 2?

f(x) = 2(3x)
On a coordinate plane, an exponential function approaches y = 0 in quadrant 2 and increases into quadrant 1. It crosses the y-axis at (0, 0.5) and goes through (2, 2).
f(x) = 3(2x)
A 2-column table has 5 rows. The first column is labeled x with entries negative 2, negative 1, 0, 1, 2. The second column is labeled f (x) with entries one-eighth, one-fourth, one-half, 1, 2.

Answer 1

1) Which exponential function has an initial value of 2?
- f(x) = 2·3^x has initial value f(0)=2·3^0 = 2.

(For comparison, f(x)=3·2^x would have initial value 3.)

2) Exponential function that crosses (0, 0.5) and (2, 2):
- a = f(0) = 0.5. Use 2 = 0.5·b^2 ⇒ b^2 = 4 ⇒ b = 2.
- So f(x) = 0.5·2^x (equivalently 2^{x-1}).

3) Table x = −2, −1, 0, 1, 2 with f(x) = 1/8, 1/4, 1/2, 1, 2:
- Ratio is ×2 each step, and f(0)=1/2, so f(x) = (1/2)·2^x (again 2^{x-1}).