I'm dealing with a problem here.
We know that two functions are the same if they have the same domain and codomain.
Let's say we have given the functions $f$ ang $g$ where $f\left(\frac{3x-8}{4}\right) = g(x)$ . The tasks says: If function f has the period 9, find the period of g. The solution in my book says that the period of g is 12.
But what I don't understand is how can two equal functions have different periods(9 and 12)?!
Can anyone help me ?
Thank you!
$f$ and $g$ are certainly not "equal functions". In fact, $g=f\circ \ell$, where $\ell$ is the linear function (with whatever domain and range) with rule $\ell(x) = (3x-8)/4 = \frac34x - 2$.
Suppose the period of $g$ is $p$. Then
$$g(x) = g(x+p)$$ $$f(\tfrac34x - 2) = f(\tfrac34(x+p) - 2)$$ $$f(\underbrace{\tfrac34x - 2}) = f(\underbrace{\tfrac34x - 2} + \tfrac34p)$$
If this is to be true for all $x$, then $\frac34p$ must be a period of $f$. The minimal such value then is that for which
$$\tfrac34p = 9$$ $$p = \tfrac43(9) = \boxed{12}$$
as desired.
