| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2013 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Moderate -0.3 Part (i) requires finding the inverse of a simple quadratic function with restricted domain—a standard textbook exercise involving rearranging y = x² + 1 and swapping variables. Part (ii) involves composing the function with itself and solving (x² + 1)² + 1 = 185/16, which requires expanding, rearranging to a quadratic in x², then taking square roots. While this is a multi-step problem worth several marks, it uses only routine algebraic techniques with no novel insight required, making it slightly easier than the average A-level question. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(x = (\pm)\sqrt{y-1}\) | B1 | OR \(y^2 = x-1\) (x/y interchange 1st) |
| \(f^{-1}: x \mapsto \sqrt{x-1}\) for \(x > 1\) | B1B1 | [3] |
| (ii) \(ff(x) = (x^2+1)^2 + 1\) | B1 | |
| \(x^2 + 1 = (\pm)3/4\) | M1 | Or \(x^4 + 2x^2 - (153/16) = 0\) |
| \(x = 3/2\) | A1 | Or \(x^2 = 9/4, (-17/4)\). www. Condone \(\pm 3/2\) |
| Alt. (ii) \(f(x) = f^{-1}(185/16) = 13/4\) | M1 | |
| \(x = f^{-1}(13/4)\) | M1 | |
| \(x = 3/2\) | A1 | Alt.(ii) \(f(3/2) = 13/4\), \(f(13/4) = 185/16\), \(x = 3/2\) |
| SC.B2 answer 1.5 with no working |
(i) $x = (\pm)\sqrt{y-1}$ | B1 | OR $y^2 = x-1$ (x/y interchange 1st)
$f^{-1}: x \mapsto \sqrt{x-1}$ for $x > 1$ | B1B1 | [3]
(ii) $ff(x) = (x^2+1)^2 + 1$ | B1 |
$x^2 + 1 = (\pm)3/4$ | M1 | Or $x^4 + 2x^2 - (153/16) = 0$
$x = 3/2$ | A1 | Or $x^2 = 9/4, (-17/4)$. www. Condone $\pm 3/2$ | [3]
Alt. (ii) $f(x) = f^{-1}(185/16) = 13/4$ | M1 |
$x = f^{-1}(13/4)$ | M1 |
$x = 3/2$ | A1 | Alt.(ii) $f(3/2) = 13/4$, $f(13/4) = 185/16$, $x = 3/2$ | B1, B1, B1
SC.B2 answer 1.5 with no working |5 The function f is defined by
$$\mathrm { f } : x \mapsto x ^ { 2 } + 1 \text { for } x \geqslant 0$$
(i) Define in a similar way the inverse function $\mathrm { f } ^ { - 1 }$.\\
(ii) Solve the equation $\operatorname { ff } ( x ) = \frac { 185 } { 16 }$.
\hfill \mbox{\textit{CAIE P1 2013 Q5 [6]}}