Question

F(x)=ax+b/2 then f^-1 (x-1) =2x-5/2 find a and b​

Answer 1

From the given inverse function, we have

$f^{-1}(x-1) = 2x - \dfrac52 \\\\ \implies f^{-1}(x) = f^{-1}((x+1)-1) = 2(x+1) - \dfrac52 = 2x-\dfrac12$

By definition of inverse function,

$f\left(f^{-1}(x)\right) = x$

so that

$f\left(2x-\dfrac12\right) = x$

$a\left(2x-\dfrac12\right) + \dfrac b2 = x$

$2ax - \dfrac a2 + \dfrac b2 = x$

$\implies\begin{cases}2a=1 \\\\ -\dfrac a2+\dfrac b2 = 0\end{cases}$

$\implies \boxed{a=\dfrac12}$

$\implies \dfrac b2 = \dfrac a2 \implies \boxed{b=\dfrac12}$