How to prove equation with functions is true?

Question

How do you prove if $f'(x+1) = f(x)$ then $f'(x) = f(x-1)$? I get that it says that when the input to $f'$ is one greater than the input to $f$ then the output of the two functions are equal. How do you prove this more formally? Also if $f'(x+1) = f(x)$, then the graph of $f'$ is the graph of $f$ shifted one unit to the right, correct? And therefore $f(x-1)$ is the graph of $f$ shifted one unit to the right, too?

Answer 1

Take $x=t-1$. Then $f'(x+1)=f'(t-1+1)=\color{blue}{f'(t)}$ is equal to $f(x)=\color{blue}{f(t-1)}$. This holds for all $t$ (as long as $f'(x+1)$ and $f(x)$ are defined). Since $x$ is just a placeholder, you can go ahead and switch $t$ with $x$ now, and you see $f'(x)=f(x-1)$.