| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modulus function |
| Type | Solve |f(x)| compared to |g(x)| with parameters: sketch then solve |
| Difficulty | Standard +0.3 This is a slightly above-average C3 question involving modulus functions and composite functions. Part (a) requires sketching a V-shaped graph and a linear function (routine). Part (b) involves solving |x-a|+a = 4x+a algebraically by considering cases (standard technique). Parts (c) and (d) test composition of functions with modulus, requiring careful substitution and case analysis. While it requires multiple techniques and careful algebraic manipulation, these are all standard C3 skills without requiring novel insight or particularly complex reasoning. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02n Sketch curves: simple equations including polynomials1.02v Inverse and composite functions: graphs and conditions for existence |
The functions f and g are defined by
$$\text{f}: x \alpha |x - a| + a, \quad x \in \mathbb{R},$$
$$\text{g}: x \alpha 4x + a, \quad x \in \mathbb{R}.$$
where $a$ is a positive constant.
\begin{enumerate}[label=(\alph*)]
\item On the same diagram, sketch the graphs of f and g, showing clearly the coordinates of any points at which your graphs meet the axes. [5]
\item Use algebra to find, in terms of $a$, the coordinates of the point at which the graphs of f and g intersect. [3]
\item Find an expression for fg(x). [2]
\item Solve, for $x$ in terms of $a$, the equation
$$\text{fg}(x) = 3a.$$ [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q5 [13]}}