| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2011 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question on basic function operations. Parts (i), (ii), and (iv) involve routine procedures: composition evaluation, sketching linear functions with their inverses, and finding inverses of simple functions (linear and cubic). Part (iii) requires differentiation and recognizing that a positive derivative implies monotonicity, hence invertibility—a standard textbook exercise. All techniques are direct applications with no problem-solving insight required. |
| Spec | 1.02m Graphs of functions: difference between plotting and sketching1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence1.07i Differentiate x^n: for rational n and sums |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(fg(2) = f(10) = 26\) | M1 A1 | Must use \(g\) first, then \(f\). co |
| \(f^{-1}(x)\) | [2] | |
| (ii) | B1 B1 B1 | \(y = f(x)\) correct in 1st, 4th quadrants. |
| \(y = f^{-1}(x)\) correct in 1st, 2nd quadrants. | ||
| \(y = x\) marked, or quoted. | ||
| [3] | ||
| (iii) \(g'(x) = 6(x-1)^2\) | B1 | co |
| \(g'(x) > \rightarrow\) no turning points | B1 √ | allow only for incorrect "6" |
| \(\rightarrow g\) is \(1 : 1\), \(g\) has an inverse. | B1 √ | following from incorrect "6" |
| [3] | ||
| (iv) \(f^{-1}(x) = \frac{x+4}{3}\) | B1 | co |
| Attempt at making \(x\) | M1 | May change \(x\) and \(y\) first. |
| Order correct. \(8, -2, \sqrt{ }, +1\) | M1 | Must all be correct, but allow for \(+8, -1\) |
| \(g^{-1}(x) = \sqrt{\frac{x-8}{2}} + 1\) | A1 | co as function of \(x\), not \(y\). |
| [4] |
$f : x \mapsto 3x - 4$ and $g : x \mapsto 2(x-1)^2 + 8$
(i) $fg(2) = f(10) = 26$ | M1 A1 | Must use $g$ first, then $f$. co
$f^{-1}(x)$ | [2] |
(ii) | B1 B1 B1 | $y = f(x)$ correct in 1st, 4th quadrants.
| | $y = f^{-1}(x)$ correct in 1st, 2nd quadrants.
| | $y = x$ marked, or quoted.
| [3] |
(iii) $g'(x) = 6(x-1)^2$ | B1 | co
$g'(x) > \rightarrow$ no turning points | B1 √ | allow only for incorrect "6"
$\rightarrow g$ is $1 : 1$, $g$ has an inverse. | B1 √ | following from incorrect "6"
| [3] |
(iv) $f^{-1}(x) = \frac{x+4}{3}$ | B1 | co
Attempt at making $x$ | M1 | May change $x$ and $y$ first.
Order correct. $8, -2, \sqrt{ }, +1$ | M1 | Must all be correct, but allow for $+8, -1$
$g^{-1}(x) = \sqrt{\frac{x-8}{2}} + 1$ | A1 | co as function of $x$, not $y$.
| [4] |10 Functions $f$ and $g$ are defined by
$$\begin{aligned}
& \mathrm { f } : x \mapsto 3 x - 4 , \quad x \in \mathbb { R } , \\
& \mathrm {~g} : x \mapsto 2 ( x - 1 ) ^ { 3 } + 8 , \quad x > 1 .
\end{aligned}$$
(i) Evaluate fg(2).\\
(ii) Sketch in a single diagram the graphs of $y = \mathrm { f } ( x )$ and $y = \mathrm { f } ^ { - 1 } ( x )$, making clear the relationship between the graphs.\\
(iii) Obtain an expression for $\mathrm { g } ^ { \prime } ( x )$ and use your answer to explain why g has an inverse.\\
(iv) Express each of $\mathrm { f } ^ { - 1 } ( x )$ and $\mathrm { g } ^ { - 1 } ( x )$ in terms of $x$.
\hfill \mbox{\textit{CAIE P1 2011 Q10 [12]}}