Easy -1.2 This is a straightforward composite function question requiring substitution of a linear function into a quadratic, followed by routine algebraic manipulation to complete the square. It involves only basic techniques with no problem-solving insight needed, making it easier than average for A-level.
3 Functions f and g are defined for \(x \in \mathbb { R }\) by
$$\begin{aligned}
& \mathrm { f } : x \mapsto 2 x + 3 \\
& \mathrm {~g} : x \mapsto x ^ { 2 } - 2 x
\end{aligned}$$
Express \(\operatorname { gf } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
Allow all these as \(\sqrt{}\) for either fg or gf
## Question 3:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f: x \mapsto 2x+3$, $g: x \mapsto x^2-2x$ | | |
| $gf(x) = (2x+3)^2 - 2(2x+3)$ | $M1$ | Must be f into g, not g into f |
| $= 4x^2 + 8x + 3$ | $A1$ | co |
| $= 4(x+1)^2 - 1$ | $3 \times B1\sqrt{}$ | Allow all these as $\sqrt{}$ for either fg or gf |
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3 Functions f and g are defined for $x \in \mathbb { R }$ by
$$\begin{aligned}
& \mathrm { f } : x \mapsto 2 x + 3 \\
& \mathrm {~g} : x \mapsto x ^ { 2 } - 2 x
\end{aligned}$$
Express $\operatorname { gf } ( x )$ in the form $a ( x + b ) ^ { 2 } + c$, where $a , b$ and $c$ are constants.
\hfill \mbox{\textit{CAIE P1 2010 Q3 [5]}}