The function is given by

MATHEMATICS · Single correct · A Level · Mathematics · Algebra and Functions · Algebra and Functions

Problem

The function

f:\mathbb R\to\mathbb R

is given by

f(x)=x^{2}

. A relation

g

is formed by interchanging

x

and

y

y=f(x)

, yielding

g(x)=\sqrt{x}

. Which statement correctly explains why

f\circ g

does NOT equal the identity on

\mathbb R

? (a) Because

f

is not one–one on

\mathbb R

, its inverse is not a function, so

f\circ g

cannot be the identity. (b) Because

g

is defined only for

x\ge 0

, the composition

f\circ g

is defined only for

x\ge 0

and thus is not the identity on all of

\mathbb R

. (c) Because taking the square root gives two possible values

\pm\sqrt{x}

g

is not single-valued, so the composition cannot recover

x

. (d) Because

f(g(x))=(\sqrt{x})^{2}=|x|

, the composition equals the absolute value of

x

, not

x

itself.