Question

How does f(x) = 10^x change over the interval from x = 5 to x = 6?
Responses
A f(x) increases by 100%f(x) increases by 100%
B f(x) increases by 500%f(x) increases by 500%
C f(x) increases by 900%f(x) increases by 900%
D f(x) increases by 1000%

Answer 1

To evaluate how the function \( f(x) = 10^x \) changes over the interval from \( x = 5 \) to \( x = 6 \), we first calculate the values of the function at these two points:

1. When \( x = 5 \):
\[
f(5) = 10^5 = 100000
\]

2. When \( x = 6 \):
\[
f(6) = 10^6 = 1000000
\]

Next, we find the change in the function over this interval:
\[
\Delta f = f(6) - f(5) = 1000000 - 100000 = 900000
\]

Now, to find the percentage increase, we use the formula for percentage change:
\[
\text{Percentage Change} = \left( \frac{\Delta f}{f(5)} \right) \times 100\%
\]
Substituting the values we found:
\[
\text{Percentage Change} = \left( \frac{900000}{100000} \right) \times 100\% = 9 \times 100\% = 900\%
\]

Thus, the function \( f(x) = 10^x \) increases by **900%** over the interval from \( x = 5 \) to \( x = 6 \).

The correct response is:
**C f(x) increases by 900%**.