When I reflect a function over the $y$ axis:
$$f(-x)$$
Consider the function $f(x+3)$. When reflecting this over the y axis:
$$f(-x+3)$$
I cannot intuitively understand why the reflection is not $f(-x-3)$? Why do we only change the sign of $x$ and not the $3$. I'm extremely confused.
Could someone explain intuitively how reflections of functions work?
For any function $g(x) = \text{ expression}$, you evaluate $g(1)$ by replacing $x$ with $1$ in the $\text{ expression}$. You can do this for any number instead of $1$.
Suppose I say $\text{ expression} = f(x+3)$. So, $g(x) = f(x+3)$. The reflection of $g(x)$ is the same as the reflection of $f(x+3)$. Reflection of $g(x)$ over the $y$-axis is $g(-x)$. Now, replace $x$ with $-x$ in the $\text{ expression}$ to find $g(-x)$ in terms of $f$. We get $g(\color{blue}{-x}) = f(\color{blue}{-x}+3)$. That's your answer.
