To determine which exponential function matches the given table of values, we need to check each function against the [tex]\( (x, f(x)) \)[/tex] pairs provided.
The pairs are:
- [tex]\( (-2, 16) \)[/tex]
- [tex]\( (-1, 8) \)[/tex]
- [tex]\( (0, 4) \)[/tex]
- [tex]\( (1, 2) \)[/tex]
- [tex]\( (2, 1) \)[/tex]
Let's analyze each function one by one to see which one fits the data:
1. [tex]\( f(x) = \frac{1}{2} (4)^x \)[/tex]
- [tex]\( f(-2) = \frac{1}{2} (4)^{-2} = \frac{1}{2} \cdot \frac{1}{16} = \frac{1}{32} \)[/tex] (does not match 16)
- Therefore, function 1 is not the correct function.
2. [tex]\( f(x) = 4 (4)^x \)[/tex]
- [tex]\( f(-2) = 4 (4)^{-2} = 4 \cdot \frac{1}{16} = \frac{1}{4} \)[/tex] (does not match 16)
- Therefore, function 2 is not the correct function.
3. [tex]\( f(x) = 4 \left(\frac{1}{2}\right)^x \)[/tex]
- [tex]\( f(-2) = 4 \left(\frac{1}{2}\right)^{-2} = 4 \cdot 4 = 16 \)[/tex] (matches 16)
- [tex]\( f(-1) = 4 \left(\frac{1}{2}\right)^{-1} = 4 \cdot 2 = 8 \)[/tex] (matches 8)
- [tex]\( f(0) = 4 \left(\frac{1}{2}\right)^x = 4 \cdot 1 = 4 \)[/tex] (matches 4)
- [tex]\( f(1) = 4 \left(\frac{1}{2}\right)^1 = 4 \cdot \frac{1}{2} = 2 \)[/tex] (matches 2)
- [tex]\( f(2) = 4 \left(\frac{1}{2}\right)^2 = 4 \cdot \frac{1}{4} = 1 \)[/tex] (matches 1)
This function matches all the given values in the table.
4. [tex]\( f(x) = \frac{1}{2} \left(\frac{1}{2}\right)^x \)[/tex]
- [tex]\( f(-2) = \frac{1}{2} \left(\frac{1}{2}\right)^{-2} = \frac{1}{2} \cdot 4 = 2 \)[/tex] (does not match 16)
- Therefore, function 4 is not the correct function.
Thus, the exponential function that correctly represents the values in the table is:
[tex]\[ f(x) = 4 \left(\frac{1}{2}\right)^x \][/tex]
