In a video dealing with the Chain Rule (Highlights Of Calculus, MIT video lectures) , Pr. Strang takes as example the $$g(x)= \sin(3x).$$
He notes that the coeffcient $3$ produces a " shrink" for the graph of $\sin(x)$ function.
The explanation he gives is that function $g$ goes , so to say, " 3 times faster".
The explantion is intuitive: I understand in which way function $g$ starting at $0$ for $x=0$ goes back again to $0$ three times sooner than the $\sin(x)$ function, that is , $g(x)=0$ for $x = \pi/3$ (for $\pi/3$ takes " in advance" the value of its triple in the $\sin(x)$ function).
However, still at the intuitive level, the image of a race between $\sin(x)$ and $\sin(3x)$ leads me to thinking that what should result is a translation $3$ units to the left.
Any graphing calculator shows that this intuition is wrong and that a shrink actually occurs.
Can this be shown rigorously?
Take $f(x)=\sin(x)$. Then $g(x)=\sin(3x)=f(3x)$. So what was at $3x$ is now at $x$. If we let $y=3x$, then the value at $y$ would be moved to $\frac{y}{3}$.