To determine which exponential function is represented by the given table, we need to verify if the provided function matches the values in the table. The table values are:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-2 & 0.2 \\
\hline
-1 & 0.4 \\
\hline
0 & 0.8 \\
\hline
1 & 1.6 \\
\hline
2 & 3.2 \\
\hline
\end{array}
\][/tex]
We are going to test two possible functions:
1. [tex]\( f(x) = 2(2^x) \)[/tex]
2. [tex]\( f(x) = 0.8(2^x) \)[/tex]
### Testing [tex]\( f(x) = 2(2^x) \)[/tex]
We will calculate [tex]\( f(x) \)[/tex] for each [tex]\( x \)[/tex] value to see if they match the table:
- For [tex]\( x = -2 \)[/tex]:
[tex]\[
f(-2) = 2(2^{-2}) = 2 \left(\frac{1}{4}\right) = \frac{2}{4} = 0.5
\][/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\[
f(-1) = 2(2^{-1}) = 2 \left(\frac{1}{2}\right) = 1
\][/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 2(2^0) = 2 \cdot 1 = 2
\][/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 2(2^1) = 2 \cdot 2 = 4
\][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 2(2^2) = 2 \cdot 4 = 8
\][/tex]
So, the resulting values are: [tex]\([0.5, 1, 2, 4, 8]\)[/tex], which do not match the provided table values ([0.2, 0.4, 0.8, 1.6, 3.2]). Hence, [tex]\( f(x) = 2(2^x) \)[/tex] is not the correct function.
### Testing [tex]\( f(x) = 0.8(2^x) \)[/tex]
Now we will calculate [tex]\( f(x) \)[/tex] for each [tex]\( x \)[/tex] value for the second function:
- For [tex]\( x = -2 \)[/tex]:
[tex]\[
f(-2) = 0.8(2^{-2}) = 0.8 \left(\frac{1}{4}\right) = 0.8 \times 0.25 = 0.2
\][/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\[
f(-1) = 0.8(2^{-1}) = 0.8 \left(\frac{1}{2}\right) = 0.8 \times 0.5 = 0.4
\][/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 0.8(2^0) = 0.8 \cdot 1 = 0.8
\][/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 0.8(2^1) = 0.8 \cdot 2 = 1.6
\][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 0.8(2^2) = 0.8 \cdot 4 = 3.2
\][/tex]
So, the resulting values are: [tex]\([0.2, 0.4, 0.8, 1.6, 3.2]\)[/tex], which perfectly match the provided table values.
Therefore, the exponential function represented by the table is:
[tex]\[ f(x) = 0.8(2^x) \][/tex]
