The functions $$$\mathrm{f}$$$ and $$$\mathrm{g}$$$ are defined by $$\[ \begin{array}{ll} \mathrm{f}(x)=x^{2}+3 & \text { for } x> 0 \\ \mathrm{~g}(x)=2 x+1 & \text { for } x> -\frac{1}{2} \end{array} \]$$ Solve the equation $$$\operatorname{fg}(x)-3=\operatorname{gf}(x)$$$. .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................

Mathematics

IGCSE&ALevel

CAIE

Exam No：9709_w20_qp_11 Year：2020 Question No：11(c)

Answer:

$\operatorname{gf}(x)=2\left(x^{2}+3\right)+1$
$(2 x+1)^{2}+3-3=2\left(x^{2}+3\right)+1 \rightarrow 2 x^{2}+4 x-6(=0)$
$(2)(x+3)(x-1)(=0)$
$x=1$

1.1.3 solve quadratic equations, and quadratic inequalities, in one unknown (By factorising, completing the square and using the formula.)

1.2.2 identify the range of a given function in simple cases, and find the composition of two given functions