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 analyze the function \( f(x) = \frac{1}{9} \left( \frac{2}{3} \right)^x \) and observe how it changes as \( x \) increases by 1 unit.

Calculating the values of \( f(x) \) for the given \( x \) values:

- For \( x = -2 \):
\[
f(-2) = \frac{1}{9} \left( \frac{2}{3} \right)^{-2} = \frac{1}{9} \cdot \frac{9}{4} = \frac{1}{4}
\]

- For \( x = -1 \):
\[
f(-1) = \frac{1}{9} \left( \frac{2}{3} \right)^{-1} = \frac{1}{9} \cdot \frac{3}{2} = \frac{1}{6}
\]

- For \( x = 0 \):
\[
f(0) = \frac{1}{9} \cdot 1 = \frac{1}{9}
\]

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

- For \( x = 2 \):
\[
f(2) = \frac{1}{9} \cdot \left( \frac{2}{3} \right)^2 = \frac{1}{9} \cdot \frac{4}{9} = \frac{4}{81}
\]

Now, let's look at how \( f(x) \) changes as \( x \) increases by 1:

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

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

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

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

As we can see from the calculations above, the amount by which \( f(x) \) changes decreases as \( x \) increases in each step. However, we also notice that the changes are proportional to \( f(x) \) at each step, which is characteristic of exponential decay.

Now we can fill in the blanks:

When the x-value changes by 1 unit, the function's value changes by **a decreasing amount**. So, \( f(x) \) demonstrates that **exponential** functions change by equal **percentages** over equal intervals.

This fits the context of exponential functions where the outputs decrease at a consistent rate relative to their current value.