WAEC Further Mathematics 2023 Paper

Concept:The inverse of a function swaps the roles of $x$ and $y$, so we solve the equation for $x$ in terms of $y$, then replace $y$ by $x$.Explanation:Let $y = f(x) = \frac{x+2}{x-3}$.Since $x \neq 3$, the denominator is not zero.Multiply both sides: $y(x-3) = x + 2$.Expand: $xy - 3y = x + 2$.Bring $x$ terms together: $xy - x = 3y + 2$.Factor out $x$: $x(y - 1) = 3y + 2$.Divide by $(y - 1)$, with $y \neq 1$: $x = \frac{3y + 2}{y - 1}$.Interchange $x$ and $y$ to get the inverse: $f^{-1}(x) = \frac{3x + 2}{x - 1}$.Thus the inverse is defined for $x \neq 1$.Answer:$f^{-1}(x) = \frac{3x + 2}{x - 1}, \quad x \neq 1$Therefore, the correct option is D.