Reflecting functions

A reflection across the $y$-axis mirrors a graph from left to right.
You form it by replacing $x$ with $-x$, giving $f(-x)$.
Each point $(x, y)$ moves to $(-x, y)$, keeping the same height.

The figure shows a function and its reflection across the $y$-axis.
Each point swaps to the opposite side while keeping its height.
The $y$-axis acts like a mirror between the two curves.

Two functions are symmetric about the $y$-axis when one is the other with $x$ replaced by $-x$.
In symbols, $h(x) = p(-x)$.
So if $h(x) = x$, its mirror image is $p(x) = -x$.

This is different from reflecting across the $x$-axis.
A $y$-axis reflection changes the input, $f(-x)$, flipping the graph left-right.
An $x$-axis reflection changes the output, $-f(x)$, flipping the graph up-down.

Reflect $f(x) = 5^x$ across the $y$-axis. What is the new function?
Replace $x$ with $-x$: $f(-x) = 5^{-x}$.
So the reflection is $5^{-x}$.

A point $(3, 5)$ lies on a graph. Where does it go when the graph is reflected across the $y$-axis?
A $y$-axis reflection sends $(x, y)$ to $(-x, y)$.
So $(3, 5)$ moves to $(-3, 5)$.

If $h(x) = x$ and $p$ is its reflection across the $y$-axis, what is $p(x)$?
Symmetry about the $y$-axis means $h(x) = p(-x)$, so $p(-x) = x$.
Replacing $-x$ with $x$ gives $p(x) = -x$.