(i)     Sketch the set $X \times Y$ in the Cartesian plane.

(ii)     Sketch the set $Y \times X$ in the Cartesian plane.

(iii)     State $(X \times Y) \cap (Y \times X)$.

Consider the function $f:X \times Y \to \mathbb{R}$ defined by $f(x,{\text{ }}y) = x + y$ and the function $g:X \times Y \to \mathbb{R}$ defined by $g(x,{\text{ }}y) = xy$.

(i)     Find the range of the function f.

(ii)     Find the range of the function g.

(iii)     Show that $f$ is an injection.

(iv)     Find ${f^{ - 1}}(\pi )$, expressing your answer in exact form.

(v)     Find all solutions to $g(x,{\text{ }}y) = \frac{1}{2}$.

(i)     

correct horizontal lines     A1

correctly labelled axes     A1

clear indication that the endpoints are not included     A1

(ii)     

fully correct diagram     A1

Note:     Do not penalize the inclusion of endpoints twice.

(iii)     the intersection is empty     A1

[5 marks]

(i)     range $(f) = \left] {0,{\text{ 1}}} \right[ \cup \left] {1,{\text{ 2}}} \right[ \cup \rm{L} \cup \left] {5,{\text{ 6}}} \right[$     A1A1

Note:     A1 for six intervals and A1 for fully correct notation.

     Accept $0 < x < 6,{\text{ }}x \ne 0{\text{, 1, 2, 3, 4, 5, 6}}$.

(ii)     range $(g) = \left[ {0,{\text{ 5}}} \right[$     A1

(iii)     Attempt at solving

$f({x_1},{\text{ }}{y_1}) = f({x_2},{\text{ }}{y_2})$     M1

$f(x,{\text{ }}y) \in \left] {y,{\text{ }}y + 1} \right[ \Rightarrow {y_1} = {y_2}$     M1

and then ${x_1} = {x_2}$     A1

so $f$ is injective     AG

(iv)     ${f^{ - 1}}(\pi ) = (\pi  - 3,{\text{ }}3)$     A1A1

(v)     solutions: (0.5, 1), (0.25, 2), $\left( {\frac{1}{6},{\text{ 3}}} \right)$, (0.125, 4), (0.1, 5)     A2

Note:     A2 for all correct, A1 for 2 correct.

[10 marks]