# A calculus of the absurd

##### 4.7.2 X-axis transformations

To transform a function $$f(x)$$ in the X-axis, we just evaluate $$f(g(x))$$, where $$g(x)$$ is a function which maps values from the $$x$$-$$y$$ plane (i.e. the usual set of axis we plot things on) to one in the $$g(x)$$-$$y$$ plane (i.e. like the usual set of axis we plot things on, except that wherever we had $$x=a$$ (where $$a$$ stands for any number) we now want $$g(x)=a$$).

This deserves a bit of explanation. Let’s imagine that $$g(x) = x - 2$$. If we plot $$f(g(x))$$ against $$g(x)$$, we might get something like this (for this specific $$f(x)$$)

We don’t want a graph of $$f(g(x))$$ against $$g(x)$$, though! We want one of $$f(g(x))$$ against $$x$$. To do this, we need to work out how to write $$g(x)$$ in terms of $$x$$, and then work out where every point on the $$g(x)$$-axis should be on the $$x$$-axis.

As $$x - 2 = g(x)$$ if we add two to each side, we obtain that $$x = g(x) + 2$$. This means that if we shift every point on the $$g(x)$$-axis two to the right then we would have the X-axis!

Thus, the graph of $$f(g(x)) = f(x-2)$$ and looks like

We can transform the X-axis in many ways, another one is stretching the graph. For example, if we set $$g(x) = \frac {1}{2} x$$, then to work out where every point on the $$g(x)$$-axis should be on the X-axis, we first rearrange $$g(x)$$, obtaining that

$$x = 2g(x)$$

and thus we stretch (not, as commonly misconceived, squish) the graph. I try to visualise it as the graph stretching as the infinite number of points on the axis are doubled (moved twice as far away as they once were).