| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2023 |
| Session | June |
| Marks | 10 |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Standard +0.3 This is a multi-part function question covering range, inverse functions, and composition. Parts (a)-(c) are standard A-level techniques: finding range of a rational function, algebraic manipulation for inverse, and verifying a composition. Part (d) requires connecting previous parts but is straightforward once you recognize ff(x) = 7/2 is outside the range found in (a). Slightly easier than average due to the guided structure and routine methods, though the composition verification requires careful algebra. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence |
The function $f$ is defined by
$$f(x) = \frac{12x}{3x + 4} \quad x \in \mathbb{R}, x \geq 0$$
(a) Find the range of $f$. [2]
(b) Find $f^{-1}$. [3]
(c) Show, for $x \in \mathbb{R}, x \geq 0$, that
$$ff(x) = \frac{9x}{3x + 1}$$ [3]
(d) Show that $ff(x) = \frac{7}{2}$ has no solutions. [2]
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q11 [10]}}