Geometry transformation in function translation Let $f(x)=x+3$. let's say we want a new function that is a translation of $f(x)$ by $5$ units to the right. If we will denote by $x'$  the new coordinate, Then For all $x$ , $x'=x+5\,\Rightarrow\, x=x'-5$.
After substituting $x=x'-5$ in f(x) we Will obtain a new function, $f(x'-5)=x'-2$.
Then the new function is $g(x)=f(x-5)=x-2$
My question is why do we need to substitute $x=x'-5$ in $f(x)$? ( what is the logic behind it?). Does it mean we just ends up with the same function? and why once we find the new function , We change $x'$ to $x$ (is there any meaning?) i don't get it
 A: Similar questions have been answered over the years, but since I didn't find a simple picture that explains this question geometrically:

In the Cartesian plane, we've introduced a new Cartesian coordinate system with its origin at location $(x_{0}, y_{0})$. If we denote the new coordinates $(x', y')$, then the formula $(x', y') = (x - x_{0}, y - y_{0})$ transforms the location of each point $(x, y)$ in the original system to the coordinates $(x', y')$ in the new system. Particularly, $(x, y) = (x_{0}, y_{0})$ corresponds to $(x', y') = (0, 0)$.
The graph of a function is a set of points independent of a coordinate system. When we write $y = f(x)$, this equation serves as a condition that a point $(x, y)$ may or may not satisfy. Points that satisfy the equation lie on the graph.
Suppose we ask, "What equation in $(x', y')$ describes the same set of points?" Since $x = x' + x_{0}$ and $y = y' + y_{0}$, we have
$$
y = f(x)\quad\text{if and only if}\quad
y' + y_{0} = f(x' + x_{0})
$$
as conditions on the plane. Here we had to express the old coordinates in terms of the new coordinates. In fancy terms, we needed to invert our translational change of coordinates.
Moving the new origin to the left moves the graph right in the new coordinates, and moving the origin up moves the graph down in the new coordinates. That's a fundamental property of reality and how we measure it.
