2022 H2 Mathematics Paper 1 Question 6

Functions

Answers

(a)

Translate the curve y=1x{y=\frac{1}{x}} by a{a} units in the positive x-axis{x\textrm{-axis}} direction.
Scale the resulting curve by a factor of a2+k{a^2+k} parallel to the y-axis.{y\textrm{-axis}.}
Translate the resulting curve by a{a} units in the positive y-axis{y\textrm{-axis}} direction.

(b)

f−1(x)=ax+kx−a.f^{-1}(x) = \frac{ax+k}{x-a}.

(c)

f2(x)=x.{f^2(x) = x.}

(d)

f2023(1)=a+k1−a.f^{2023}(1) = \frac{a+k}{1-a}.

Full solutions

(a)

By long division,
f(x)=ax+kx−a=a+a2+kx−a\begin{align*} f(x) &= \frac{ax+k}{x-a} \\ &= a + \frac{a^2 + k}{x-a} \end{align*}
1x→1x−a→a2+kx−a→a+a2+kx−a\frac{1}{x} \to \frac{1}{x-a} \to \frac{a^2 + k}{x-a} \to a + \frac{a^2 + k}{x-a}
Translate the curve y=1x{y=\frac{1}{x}} by a{a} units in the positive x-axis{x\textrm{-axis}} direction.

Scale the resulting curve by a factor of a2+k{a^2+k} parallel to the y-axis.{y\textrm{-axis}.}

Translate the resulting curve by a{a} units in the y-axis{y\textrm{-axis}} direction. ■{\blacksquare}

(b)

Let y=ax+kx−axy−ay=ax+kxy−ax=ay+kx(y−a)=ay+kx=ay+ky−af−1(x)=ax+kx−a  ■\begin{align*} \textrm{Let } y &= \frac{ax+k}{x-a} \\ xy - ay &= ax + k \\ xy - ax &= ay + k \\ x(y-a) &= ay+k \\ x &= \frac{ay+k}{y-a} \\ f^{-1}(x) &= \frac{ax+k}{x-a} \; \blacksquare \end{align*}

(c)

We observe from part (b) that f(x)=f−1(x){f(x) = f^{-1}(x)}
f2(x)=ff(x)=ff−1(x)=x  ■\begin{align*} f^2 (x) &= ff(x) \\ &= ff^{-1}(x) \\ &= x \; \blacksquare \end{align*}

(d)

f2023(1)=f2021f2(1)=f2021(1)=⋯=f(1)=a+k1−a  ■\begin{align*} f^{2023}(1) &= f^{2021}f^2(1) \\ &= f^{2021} (1) \\ &= \cdots \\ &= f(1) \\ &= \frac{a+k}{1-a} \; \blacksquare \end{align*}

Question Commentary

Note that the two equations for part (a) are not structurally similar at all: the f(x){f(x)} expression has x{x} in the numerator while 1x{\frac{1}{x}} does not. Attempts to apply transformations in these forms will be extremely challenging.

We thus should perform long division on the f(x){f(x)} expression so the required transformation is easier to identify. The regular rule of thumb works here in deciding the order of transformations: translation before scaling/reflection for x{x} transformations, and then scaling/reflection before translation for y{y} transformations.

The rest of the question involves the idea of a "self-inverse function". This has come up numerous times in the past, but the exact technique needed to solve part (c) and (d) was last seen in 2014 paper 1 question 1 and 2009 paper 2 question 3.