Find the function $g(t)$ if $f(t)=3 t-2$ and $(g \circ f)^{-1}(t)=t-2$.

Question

Aicharya · Accepted Answer

Given, $(g \circ f)^{-1}(t)=t-2$ $\begin{aligned} & \Rightarrow \quad(g \circ f)(t)=t+2 \ & \Rightarrow \quad g(f(t))=t+2 \ & \Rightarrow \quad g(3 t-2)=(t+2) \quad\{\because f(t)=3 t-2\} \ \end{aligned}$ Now replace $t$ by $\frac{t}{3}$, we get $g(t-2)=\left(\frac{t}{3}+2\right)$ Again replace $t$ by $t+2$, we get $g(t)=\frac{t+2}{3}+2=\frac{t+8}{3}$

Find the function \(g(t)\) if \(f(t)=3 t-2\) and \((g \circ f)^{-1}(t)=t-2\).

Solution