Function Transformations Take the function $f(x)=x^2$. Apply the transformation so that $f(x)=x^2+C$. One option for the two sides of the equation to remain equivalent is for all $y$ values for which $x$ is defined to also increase by $C$. However could a second scenario not occur. Let me re-label the original function $f(x)=x_1^2$. Once again apply the transformation so that $f(x)=x_1^2+C$.Instead of there being an increase in $y$. Could $x_1$ ($x_1$ represents all original $x$ values)not decrease to another $x$ lets call it $x_2$ so that  $x_2^2+C=x_1^2$. In that way the equation would remain balanced. However what troubles me about this ( and for that matter any transformation along the $x$ axis) is that if every $x$ value has been changed and $y$ is a function of $x$ then surely every $y$ value must be changed as well. Once again I do not believe I am correct I am just not quite sure what I am doing wrong. All help is appreciated
A

One difficulty with $x_2^2 + C = x_1^2$ occurs when $x_1 = 0$ and $C > 0$.
In that case you would need $x_2^2 = -C < 0,$ which is impossible
if $x_2$ is a real number.
In addition, the mapping you would need between $x_1$ and $x_2$
to maintain $x_2^2 + C = x_1^2$ (when it works at all)
is at least as complicated as the function $f(x) = x^2 + C$ itself.
It sometimes is reasonable to transform the input to an equation rather
than the output, that is, given functions $f$ and $g$, you find
a function $h$ such that $f(h(x)) = g(f(x))$,
(which I think is what you are trying to do).
But it doesn't always work and it is often not helpful.
