Question

Suppose that f is an exponential function such that f(x) = 4 * 2^x. Which answer shows that this exponential function grows by equal factors over equal intervals?

a) f(4)÷f(3) = f(7)÷f(6)

b) f(4)+f(3) = f(7)+f(6)

c) f(4)⋅f(3) = f(7)⋅f(6)

d) f(4)−f(3) = f(7)−f(6)

Answer 1

The correct answer is a) f(4)÷f(3) = f(7)÷f(6), showing the function's growth by equal factors (specifically by a factor of 2) over equal intervals, reflecting the defining characteristic of exponential functions.

Step-by-step explanation:

The student asked which answer choice demonstrates that the exponential function f(x) = 4 * 2x grows by equal factors over equal intervals. To determine this, we need to analyze how the function's value changes between consecutive inputs.

The correct answer is a) f(4)÷f(3) = f(7)÷f(6), which shows that the function grows by equal factors over equal intervals of x. This is because the function values at these points will result in the constant factor of 2, which is the base of the exponential function. Substituting x into the function we get:

1. f(4) = 4 * 24 = 4 * 16 = 64
2. f(3) = 4 * 23 = 4 * 8 = 32
3. f(7) = 4 * 27 = 4 * 128 = 512
4. f(6) = 4 * 26 = 4 * 64 = 256

Now, dividing f(4) by f(3) yields 64/32 = 2 and similarly f(7) by f(6) gives 512/256 = 2. So, both ratios are equal, demonstrating that the growth was by a factor of 2 over those intervals. This is an essential property of exponential functions: they grow by a constant factor over equal interval steps.