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 - \frac52 \\\\ \implies f^(-1)(x) = f^(-1)((x+1)-1) = 2(x+1) - \frac52 = 2x-\frac12`

By definition of inverse function,

`f\left(f^(-1)(x)\right) = x`

so that

`f\left(2x-\frac12\right) = x`

`a\left(2x-\frac12\right) + \frac b2 = x`

`2ax - \frac a2 + \frac b2 = x`

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

`\implies \boxed{a=\frac12}`

`\implies \frac b2 = \frac a2 \implies \boxed{b=\frac12}`