Let's start by evaluating the given function [tex]\( f(x) = 49^x \)[/tex] at [tex]\( x = -\frac{1}{2} \)[/tex].
First, recall that:
[tex]\[ f(x) = 49^x \][/tex]
Given [tex]\( x = -\frac{1}{2} \)[/tex], we want to find [tex]\( f\left(-\frac{1}{2}\right) \)[/tex].
Substitute [tex]\( x = -\frac{1}{2} \)[/tex] into the function:
[tex]\[ f\left(-\frac{1}{2}\right) = 49^{-\frac{1}{2}} \][/tex]
Now, let's simplify [tex]\( 49^{-\frac{1}{2}} \)[/tex]:
1. Recognize that 49 can be written as [tex]\( 7^2 \)[/tex]:
[tex]\[ 49 = 7^2 \][/tex]
2. This allows us to rewrite the expression as:
[tex]\[ 49^{-\frac{1}{2}} = (7^2)^{-\frac{1}{2}} \][/tex]
3. Apply the power of a power property [tex]\((a^m)^n = a^{mn}\)[/tex]:
[tex]\[ (7^2)^{-\frac{1}{2}} = 7^{2 \cdot -\frac{1}{2}} = 7^{-1} \][/tex]
4. Simplifying [tex]\( 7^{-1} \)[/tex]:
[tex]\[ 7^{-1} = \frac{1}{7} \][/tex]
Therefore:
[tex]\[ f\left(-\frac{1}{2}\right) = \frac{1}{7} \][/tex]
Thus the correct option is:
C) [tex]\(\frac{1}{7}\)[/tex]
