Problem
(c) The diagram shows the function $f(x)=\frac{1}{x^{2}+1}$ for $x \geq 0$. State the domain and range of the inverse function $f^{-1}(x)$. (d) Consider the function $f(x)=x^{2}-4 x+6$. (i) Explain why the domain of $f(x)$ must be restricted if $f(x)$ is to have an inverse function. (ii) Given that the domain of $f(x)$ is restricted to $x \leq 2$, find an expression for the inverse function $f^{-1}(x)$.
Solution
Step 1 ：Rewrite the equation as \(y^2 - 4y + 6 - x = 0\) and complete the square for the \(y\) terms: \((y - 2)^2 = x - 2\)
Step 2 ：Take the negative square root and solve for \(y\) to get the inverse function: \(y = 2 - \sqrt{x - 2}\)
Step 3 ：\(\boxed{f^{-1}(x) = 2 - \sqrt{x - 2}}\)
