A systematic method for transforming piece-wise functions into one another.

Question

I need to transform $f(t)$ to $g(t)$ but I am unsure how I would go about doing that.

(ie) put: (A, B, C are arbitrary constants)

$g(t) = Af(t-B) + C$

also note: $f(t) = f(t + 2\pi ), g(t) = g(t+2)$

$f(t) =$

\begin{array}{ll} \pi & 0 < t\leq \pi \\ t & \pi \leq t< 2 \pi \\ \end{array}

$g(t) =$

\begin{array}{ll} 0 & 0 < t\leq 1 \\ t-1 & 1 \leq t < 2 \\ \end{array}

Answer 1

First of all, note that $f$ is constant in $(0,\pi]$, so is $g$ in $(0,1]$. Also, $f$ is a line in $(\pi,2\pi]$, so is $g$ in $(1,2]$. That way, it's obvious that $g(t)$ must be an affine transformation of $f(\pi t)$. Then adjust the constants missing. $$g(t)=Af(\pi t) + B$$ if $t=1/2$, we get that $$A\pi +B=0 \Longrightarrow B=-A\pi$$ and if $t=3/2$, we get that $$3A\pi/2 + (-A\pi) = 1/2$$ so we have that $A=1/\pi$ and that $B=-1$. That way, we conclude that $$g(t)=\frac{f(\pi t)}{\pi} -1$$