Question

This table shows some values of the function f(x)=
1
9
2
3
x.
x
–
2
–
1 0 1 2
f(x)
1/4
1/6
1/9
2/27
4/81
Complete the sentences.
When the x-value changes by 1 unit, the function's value changes by
. So, f(x) demonstrates that
functions change by equal
over equal intervals.

Answer 1

To complete the sentences, we first need to calculate the change in the function's value \( f(x) \) for each 1 unit change in \( x \):

- For \( x = -2 \) to \( x = -1 \):
\[
\Delta f = f(-1) - f(-2) = \frac{1}{6} - \frac{1}{4} = \frac{2}{12} - \frac{3}{12} = -\frac{1}{12}
\]

- For \( x = -1 \) to \( x = 0 \):
\[
\Delta f = f(0) - f(-1) = \frac{1}{9} - \frac{1}{6} = \frac{2}{18} - \frac{3}{18} = -\frac{1}{18}
\]

- For \( x = 0 \) to \( x = 1 \):
\[
\Delta f = f(1) - f(0) = \frac{2}{27} - \frac{1}{9} = \frac{2}{27} - \frac{3}{27} = -\frac{1}{27}
\]

- For \( x = 1 \) to \( x = 2 \):
\[
\Delta f = f(2) - f(1) = \frac{4}{81} - \frac{2}{27} = \frac{4}{81} - \frac{6}{81} = -\frac{2}{81}
\]

From these calculations, we can observe that as \( x \) increases by 1 unit, \( f(x) \) generally decreases, with the changes in \( f(x) \) appearing to get smaller as \( x \) increases.

Now, summarizing our findings and completing the sentences:

When the x-value changes by 1 unit, the function's value changes by a varying amount (decreasing each time but not consistently). So, f(x) demonstrates that **exponential** functions change by equal **proportional or fractional values** over equal intervals.

This conclusion reflects the nature of the function \( f(x) \) as it appears to exhibit characteristics of exponential decay.