Question

Let $f(x) = ax + b$ for $0 \le x \le 1$, where $a$ and $b$ are constants such that $a < 0$. What is the inverse of $f(x)$ in terms of $a$ and $b$?

Answer 1

it doesn't really matter whether a and b are negative.
If f(x) = ax+b then
f^{^-1}(x) = (x-b)/a
or, f^{^-1}(x) = 1/a x - b/a
it's easy to check, since we know that
f^{^-1}(f(x)) = f(f^{^-1}(x)) = x

f(f^{^-1}(x)) = a[(x-b)/a)+b = x
f^{^-1}(f(x)) = (f(x)-b)/a+b = (ax+b-b)/a = x