Why are function transformations inversed in equation form?

Question

I recently came across this topic in math, about function and transformations.

For example:

Let's say I have a function $f(x) = x^2$

If I want to shift the function 3 units to the right, I need to subtract the $x$-value in the equation by $3$, this is counterintuitive to the way I would think: add $3$ to the function.

The equation becomes:

$f(x-3)=(x-3)^2$

And surprisingly not $f(x+3)=(x+3)^2$

Anyone could explain to me why is this the case?

Answer 1

Compare the function $$g(x)=(x-3)^2$$ with $$f(x)=x^2$$

Note that $$ g(x+3)=f(x)$$ That is we have to move our $x$ three units to the right to get the same values out of the $g$