To determine which exponential function is represented by the table, we need to analyze the table values for [tex]\( f(x) \)[/tex] for given [tex]\( x \)[/tex].
The table is given as:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-2 & 12.5 \\
-1 & 2.5 \\
0 & 0.5 \\
1 & 0.1 \\
2 & 0.02 \\
\hline
\end{array}
\][/tex]
We are given four possible exponential functions to consider:
1. [tex]\( f(x) = 0.2 \left( 0.5^x \right) \)[/tex]
2. [tex]\( f(x) = 0.5 \left( 5^x \right) \)[/tex]
3. [tex]\( f(x) = 0.5 \left ( 0.2^x \right) \)[/tex]
4. [tex]\( f(x) = 0.2 \left( 0.2^x \right) \)[/tex]
We need to check each function to see which one corresponds with all the values in the table.
1. For [tex]\( f(x) = 0.2 \left( 0.5^x \right) \)[/tex]:
- Calculating for [tex]\( x = -2 \)[/tex]:
[tex]\[
f(-2) = 0.2 \left( 0.5^{-2} \right) = 0.2 \left( 4 \right) = 0.8
\][/tex]
- The value does not match [tex]\( f(-2) = 12.5 \)[/tex]. So this function is not the correct one.
2. For [tex]\( f(x) = 0.5 \left( 5^x \right) \)[/tex]:
- Calculating for [tex]\( x = -2 \)[/tex]:
[tex]\[
f(-2) = 0.5 \left( 5^{-2} \right) = 0.5 \left( 0.04 \right) = 0.02
\][/tex]
- This value does not match [tex]\( f(-2) = 12.5 \)[/tex]. Thus, this function is not the correct one.
3. For [tex]\( f(x) = 0.5 \left( 0.2^x \right) \)[/tex]:
- Calculating for [tex]\( x = -2 \)[/tex]:
[tex]\[
f(-2) = 0.5 \left( 0.2^{-2} \right) = 0.5 \left( 25 \right) = 12.5
\][/tex]
- Calculating for [tex]\( x = -1 \)[/tex]:
[tex]\[
f(-1) = 0.5 \left( 0.2^{-1} \right) = 0.5 \left( 5 \right) = 2.5
\][/tex]
- Calculating for [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 0.5 \left( 0.2^{0} \right) = 0.5 \left( 1 \right) = 0.5
\][/tex]
- Calculating for [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 0.5 \left( 0.2^{1} \right) = 0.5 \left( 0.2 \right) = 0.1
\][/tex]
- Calculating for [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 0.5 \left( 0.2^{2} \right) = 0.5 \left( 0.04 \right) = 0.02
\][/tex]
- All these values align perfectly with the given table values. So this function seems to be a match.
4. For [tex]\( f(x) = 0.2 \left( 0.2^x \right) \)[/tex]:
- Calculating for [tex]\( x = -2 \)[/tex]:
[tex]\[
f(-2) = 0.2 \left( 0.2^{-2} \right) = 0.2 \left( 25 \right) = 5
\][/tex]
- This value does not match [tex]\( f(-2) = 12.5 \)[/tex]. Thus, this function is not correct.
In conclusion, the exponential function that matches the table is:
[tex]\[
f(x) = 0.5 \left( 0.2^x \right)
\][/tex]
