The function $$\(\mathrm{f}\)$$ is defined by $$\(\mathrm{f}(x)=2 x^{2}+12 x+11\)$$ for $$\(x \leqslant-4\)$$. Find an expression for $$\(\mathrm{f}^{-1}(x)\)$$ and state the domain of $$\(\mathrm{f}^{-1}\)$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_m20_qp_12 Year：2020 Question No：9(b)
Answer:
\[
\begin{array}{l}
y=2(x+3)^{2}-7 \rightarrow 2(x+3)^{2}=y+7 \rightarrow(x+3)^{2}=\frac{y+7}{2}
x+3=(\pm) \sqrt{\frac{y+7}{2}} \rightarrow x=(\pm) \sqrt{\frac{y+7}{2}}-3 \rightarrow \mathrm{f}^{-1}(x)=-\sqrt{\frac{x+7}{2}}-3
\end{array}
\]
Domain: \(x \geqslant-5\) or \(\geqslant-5\) or \([-5, \infty)\)
\begin{array}{l}
y=2(x+3)^{2}-7 \rightarrow 2(x+3)^{2}=y+7 \rightarrow(x+3)^{2}=\frac{y+7}{2}
x+3=(\pm) \sqrt{\frac{y+7}{2}} \rightarrow x=(\pm) \sqrt{\frac{y+7}{2}}-3 \rightarrow \mathrm{f}^{-1}(x)=-\sqrt{\frac{x+7}{2}}-3
\end{array}
\]
Domain: \(x \geqslant-5\) or \(\geqslant-5\) or \([-5, \infty)\)
Knowledge points:
1.2.1 understand the terms function, domain, range, one-one function, inverse function and composition of functions
1.2.3 determine whether or not a given function is one-one, and find the inverse of a one-one function in simple cases
Solution:
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