Question

Part A
Find the inverse of the rational function f (x) = 2x/x+3,x ≠-3

Answer 1

If
`f^(-1)(x)` is the inverse of
`f(x)`, then

`f\left(f^(-1)(x)\right) = (2f^(-1)(x))/(f^(-1)(x)+3) = x`

Solve for
`f^(-1)(x)`. Since
`f(x)` is only defined for
`x\\eq-3`, this means that
`f\left(f^(-1)(x)\right)` is only defined for
`f^(-1)(x)\\eq-3`. Then the denominator is never zero, so we can multiply both sides by it:

`(2f^(-1)(x))/(f^(-1)(x)+3) = x \\\\ (2f^(-1)(x))/(f^(-1)(x)+3) \cdot (f^(-1)(x)+3) = x(f^(-1)(x)+3) \\\\ 2f^(-1)(x) = x(f^(-1)(x)+3) \\\\ 2f^(-1)(x) = xf^(-1)(x) + 3x \\\\ 2f^(-1)(x) - xf^(-1)(x) = 3x \\\\ (2-x)f^(-1)(x) = 3x \\\\ \boxed{f^(-1)(x) = (3x)/(2-x)}`

provided that x ≠ 2.