Find ${f^{ - 1}}(x)$.

Show that $\left( {g \circ {f^{ - 1}}} \right)(x) = \frac{5}{{x + 2}}$.

Find the $y$-intercept of the graph of $h$.

Hence, sketch the graph of $h$.

For the graph of ${h^{ - 1}}$, write down the $x$-intercept;

For the graph of ${h^{ - 1}}$, write down the equation of the vertical asymptote.

Given that ${h^{ - 1}}(a) = 3$, find the value of $a$.

interchanging $x$ and $y$     (M1)

eg     $x = 3y - 2$

${f^{ - 1}}(x) = \frac{{x + 2}}{3}{\text{   }}\left( {{\text{accept }}y = \frac{{x + 2}}{3},{\text{ }}\frac{{x + 2}}{3}} \right)$     A1     N2

[2 marks]

attempt to form composite (in any order)     (M1)

eg     $g\left( {\frac{{x + 2}}{3}} \right),{\text{ }}\frac{{\frac{5}{{3x}} + 2}}{3}$

correct substitution     A1

eg     $\frac{5}{{3\left( {\frac{{x + 2}}{3}} \right)}}$

$\left( {g \circ {f^{ - 1}}} \right)(x) = \frac{5}{{x + 2}}$     AG     N0

[2 marks]

valid approach     (M1)

eg     $h(0),{\text{ }}\frac{5}{{0 + 2}}$

$y = \frac{5}{2}{\text{   }}\left( {{\text{accept (0, 2.5)}}} \right)$     A1     N2

[2 marks]

     A1A2     N3

Notes:     Award A1 for approximately correct shape (reciprocal, decreasing, concave up).

     Only if this A1 is awarded, award A2 for all the following approximately correct features: y-intercept at $(0, 2.5)$, asymptotic to x-axis, correct domain $x \geqslant 0$.

     If only two of these features are correct, award A1.

[3 marks]

$x = \frac{5}{2}{\text{   }}\left( {{\text{accept (2.5, 0)}}} \right)$     A1     N1

[1 mark]

$x = 0$   (must be an equation)     A1     N1

[1 mark]

METHOD 1

attempt to substitute $3$ into $h$ (seen anywhere)     (M1)

eg     $h(3),{\text{ }}\frac{5}{{3 + 2}}$

correct equation     (A1)

eg     $a = \frac{5}{{3 + 2}},{\text{ }}h(3) = a$

$a = 1$     A1     N2

[3 marks]

METHOD 2

attempt to find inverse (may be seen in (d))     (M1)

eg     $x = \frac{5}{{y + 2}},{\text{ }}{h^{ - 1}} = \frac{5}{x} - 2,{\text{ }}\frac{5}{x} + 2$

correct equation, $\frac{5}{x} - 2 = 3$     (A1)

$a = 1$     A1     N2

[3 marks]