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2014
P1 Q1
Topical
Functions
14 P1 Q1
2014 H2 Mathematics Paper 1 Question 1
Functions
Answers
(ii)
f
3
(
x
)
=
x
{f^3(x)=x}
f
3
(
x
)
=
x
Full solutions
(i)
f
2
(
x
)
=
f
(
1
1
−
x
)
=
1
1
−
1
1
−
x
=
1
÷
1
−
x
−
1
1
−
x
=
1
×
1
−
x
−
x
=
x
−
1
x
\begin{align*} f^2(x) &= f \left( \frac{1}{1-x} \right) \\ &= \frac{1}{1-\frac{1}{1-x}} \\ &= 1 \div \frac{1-x-1}{1-x} \\ &= 1 \times \frac{1-x}{-x} \\ &= \frac{x-1}{x} \end{align*}
f
2
(
x
)
=
f
(
1
−
x
1
)
=
1
−
1
−
x
1
1
=
1
÷
1
−
x
1
−
x
−
1
=
1
×
−
x
1
−
x
=
x
x
−
1
y
=
1
1
−
x
y
−
x
y
=
1
x
y
=
y
−
1
x
=
y
−
1
y
\begin{gather*} y = \frac{1}{1-x} \\ y-xy = 1 \\ xy = y-1 \\ x = \frac{y-1}{y} \end{gather*}
y
=
1
−
x
1
y
−
x
y
=
1
x
y
=
y
−
1
x
=
y
y
−
1
∴
f
−
1
(
x
)
=
x
−
1
x
\therefore f^{-1}(x) = \frac{x-1}{x}
∴
f
−
1
(
x
)
=
x
x
−
1
Hence
f
2
(
x
)
=
f
−
1
(
x
)
■
{f^2(x)=f^{-1}(x) \; \blacksquare}
f
2
(
x
)
=
f
−
1
(
x
)
■
(ii)
f
3
(
x
)
=
f
f
2
(
x
)
=
f
f
−
1
(
x
)
=
x
■
\begin{align*} f^3(x) &= ff^2(x) \\ &= ff^{-1}(x) \\ &= x \; \blacksquare \end{align*}
f
3
(
x
)
=
f
f
2
(
x
)
=
f
f
−
1
(
x
)
=
x
■
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