(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 $f^{-1}(x)$.

Answer

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Answer

$\boxed{f^{-1}(x) = 2 - \sqrt{x - 2}}$

Steps

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 ：$\boxed{f^{-1}(x) = 2 - \sqrt{x - 2}}$