2014 A-level H2 Mathematics (9740) Paper 1 Question 1 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

(i)
f^{2}(x)
=f[f(x)]
=f(\frac{1}{1-x})
=\frac{1}{1-\frac{1}{1-x}}
=\frac{1}{\frac{1-x-1}{1-x}}
=\frac{x-1}{x}. x\neq{0}, x\neq{1}

let y=f(x)
y(1-x)=1
x=\frac{y-1}{y}
f^{-1}(x)=\frac{x-1}{x}, x\neq{1}, x\neq{0}

Since D_{f^{-1}}=R_{f}=\mathbb{R} \backslash \{0,1\}

We have that f^{2}(x)=f^{-1}(x)

(ii)
f^{2}(x)=f^{-1}(x)
f[f^{2}(x)]=f[f^{-1}(x)]
f^{3}(x)=x, x\neq{1}, x\neq{0}

Personal Comments:
Students must understand what it means for two functions to be equal. It isn’t just about the rule (loosely put, formula) the same, but they require the same domain, and as a result the same range too. Stating the domains in (i) is important to show true understanding of what equal functions mean. It is saddening that some students also thought that f^{2}(x)=f(x)f(x) which is not the case. f^{2}(x) is a composite function, “f in f” while f(x)f(x) is a product of two functions.

For (ii), students just need to be able to manipulate composite functions. It is always good habit to state domains for functions as they are meaningless without a domain.

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