2012 H2 Mathematics Paper 1 Question 7

Functions

Answers

{y=x}

Translate the curve ${y=\frac{1}{x}}$ by ${1}$ unit in the positive ${x\textrm{-}}$ axis direction.
Scale the resulting curve with scale factor ${k+1}$ unit parallel to the ${y\textrm{-}}$ axis.
Translate the resulting curve by ${1}$ unit in the positive ${y\textrm{-}}$ axis direction.

\begin{gather*} y = \frac{x+k}{x-1} \\ xy-y = x + k \\ xy-x = y+k \\ x(y-1) = y + k \\ x = \frac{y+k}{y-1} \end{gather*}

g^{-1}(x) = \frac{x+k}{x-1}

Hence

{g(x)=g^{-1}(x)}

for all

{x}

in the domain of

{g}

(since

D_{g^{-1}} = R_g = D_g = \mathbb{R}\setminus\{1\}

).
Hence

{g}

is self-inverse

{\blacksquare}

Line of symmetry of the curve:

y=x \; \blacksquare

By long division,

g(x)=1+\frac{k+1}{x-1}

Sequence of transformations required:

Translate the curve ${y=\frac{1}{x}}$ by ${1}$ unit in the positive ${x\textrm{-}}$ axis direction.
Scale the resulting curve with scale factor ${k+1}$ unit parallel to the ${y\textrm{-}}$ axis.
Translate the resulting curve by ${1}$ unit in the positive ${y\textrm{-}}$ axis direction.