The function $$\(\mathrm{g}\)$$ is defined by $$\(\mathrm{g}(x)=2 x-3\)$$ for $$\(x \leqslant k\)$$. For the case where $$\(k=-1\)$$, solve the equation $$\(\mathrm{fg}(x)=193\)$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_m20_qp_12 Year：2020 Question No：9(c)
Answer:
\[
\begin{array}{l}
f g(x)=8 x^{2}-7 \\
8 x^{2}-7=193 \rightarrow x^{2}=25 \rightarrow x=-5 \text { only }
\end{array}
\]
Alternative method for question 9(c)
\[
\begin{array}{l}
\mathrm{g}(x)=\mathrm{f}^{-1}(193) \rightarrow 2 x-3=-\sqrt{100}-3 \\
x=-5 \text { only }
\end{array}
\]
\begin{array}{l}
f g(x)=8 x^{2}-7 \\
8 x^{2}-7=193 \rightarrow x^{2}=25 \rightarrow x=-5 \text { only }
\end{array}
\]
Alternative method for question 9(c)
\[
\begin{array}{l}
\mathrm{g}(x)=\mathrm{f}^{-1}(193) \rightarrow 2 x-3=-\sqrt{100}-3 \\
x=-5 \text { only }
\end{array}
\]
Knowledge points:
1.1.3 solve quadratic equations, and quadratic inequalities, in one unknown (By factorising, completing the square and using the formula.)
1.2.1 understand the terms function, domain, range, one-one function, inverse function and composition of functions
1.2.2 identify the range of a given function in simple cases, and find the composition of two given functions
Solution:
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