Standard +0.3 This is a comprehensive but routine question covering standard AS-level techniques: completing the square, finding roots via quadratic formula, identifying turning points, and understanding inverse function existence. While multi-part with several marks, each component is a textbook exercise requiring no novel insight—slightly easier than average due to straightforward application of well-practiced methods.
The diagram below shows a sketch of the graph of \(y = f ( x )\), where
$$f ( x ) = 2 x ^ { 2 } + 12 x + 10 .$$
The graph intersects the \(x\)-axis at the points \(( p , 0 ) , ( q , 0 )\) and the \(y\)-axis at the point \(( 0,10 )\).
\includegraphics[max width=\textwidth, alt={}, center]{72bb1603-edbd-4e2e-bf2b-f33bb667e61b-5_1004_1171_648_440}
a) Write down the value of \(f f ( p )\).
b) Determine the values of \(p\) and \(q\).
c) Express \(f ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b , c\) are constants whose values are to be found. Write down the coordinates of the minimum point.
d) Explain why \(f ^ { - 1 } ( x )\) does not exist.
e) The function \(g ( x )\) is defined as
$$g ( x ) = f ( x ) \quad \text { for } \quad - 3 \leqslant x < \infty .$$
i) Find an expression for \(g ^ { - 1 } ( x )\).
ii) Sketch the graph of \(y = g ^ { - 1 } ( x )\), indicating the coordinates of the points where the graph intersects the \(x\)-axis and the \(y\)-axis.
The diagram below shows a sketch of the graph of $y = f ( x )$, where
$$f ( x ) = 2 x ^ { 2 } + 12 x + 10 .$$
The graph intersects the $x$-axis at the points $( p , 0 ) , ( q , 0 )$ and the $y$-axis at the point $( 0,10 )$.\\
\includegraphics[max width=\textwidth, alt={}, center]{72bb1603-edbd-4e2e-bf2b-f33bb667e61b-5_1004_1171_648_440}\\
a) Write down the value of $f f ( p )$.\\
b) Determine the values of $p$ and $q$.\\
c) Express $f ( x )$ in the form $a ( x + b ) ^ { 2 } + c$, where $a , b , c$ are constants whose values are to be found. Write down the coordinates of the minimum point.\\
d) Explain why $f ^ { - 1 } ( x )$ does not exist.\\
e) The function $g ( x )$ is defined as
$$g ( x ) = f ( x ) \quad \text { for } \quad - 3 \leqslant x < \infty .$$
i) Find an expression for $g ^ { - 1 } ( x )$.\\
ii) Sketch the graph of $y = g ^ { - 1 } ( x )$, indicating the coordinates of the points where the graph intersects the $x$-axis and the $y$-axis.
\hfill \mbox{\textit{WJEC Unit 3 2022 Q12}}