Answer :
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]
[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]
