Question

How does f(x) = 8^x
change over the interval from x = 1 to x = 2?
Responses
A f(x) decreases by a factor of 1f(x) decreases by a factor of 1
B f(x) increases by a factor of 1f(x) increases by a factor of 1
C f(x) decreases by a factor of 8f(x) decreases by a factor of 8
D f(x) increases by a factor of 8

Answer 1

To understand how \( f(x) = 8^x \) changes over the interval from \( x = 1 \) to \( x = 2 \), we need to evaluate the function at both of these points.

1. Calculate \( f(1) \):
\[
f(1) = 8^1 = 8
\]

2. Calculate \( f(2) \):
\[
f(2) = 8^2 = 64
\]

Now, we can determine how \( f(x) \) changes from \( f(1) \) to \( f(2) \):
- Starting from \( f(1) = 8 \) and going to \( f(2) = 64 \), we can find the factor of increase.

To find the factor of increase, we divide \( f(2) \) by \( f(1) \):
\[
\text{Factor of Increase} = \frac{f(2)}{f(1)} = \frac{64}{8} = 8
\]

Thus, \( f(x) \) increases by a factor of 8 from \( x = 1 \) to \( x = 2 \).

The correct response is:
**D** \( f(x) \) increases by a factor of 8.