To find the [tex]$y$[/tex]-intercept of the function [tex]\( f(x) = \left( \frac{1}{2} \right)^x \)[/tex], you need to determine the point where the graph of the function crosses the [tex]$y$[/tex]-axis. This happens when [tex]\( x = 0 \)[/tex].
1. Substitute [tex]\( x = 0 \)[/tex] into the function [tex]\( f(x) = \left( \frac{1}{2} \right)^x \)[/tex]:
[tex]\[
f(0) = \left( \frac{1}{2} \right)^0
\][/tex]
2. Recall that any non-zero number raised to the power of 0 is equal to 1:
[tex]\[
\left( \frac{1}{2} \right)^0 = 1
\][/tex]
3. Therefore, [tex]\( f(0) = 1 \)[/tex], so the [tex]$y$[/tex]-intercept is at the point where [tex]\( x = 0 \)[/tex] and [tex]\( y = 1 \)[/tex]. This point is written as:
[tex]\[
(0, 1)
\][/tex]
Hence, the [tex]$y$[/tex]-intercept of the function [tex]\( f(x) = \left( \frac{1}{2} \right)^x \)[/tex] is [tex]\(\boxed{(0, 1)}\)[/tex].
The correct answer is:
C. [tex]\((0, 1)\)[/tex]
