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!

  • 1
    $\begingroup$ The functions aren't equal. They take on all the same values, but for different inputs. like sinx, cosx, and sin(2x) $\endgroup$ – genisage Sep 16 '14 at 20:54

$f$ 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.


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