Let be defined by and let be defined by (principal square root)

MATHEMATICS · Single correct · CBSE · Mathematics · Relations and Functions · Relations and Functions

Problem

Let

f:\mathbb R\to[0,\infty)

be defined by

f(x)=x^{2}

and let

g:[0,\infty)\to[0,\infty)

be defined by

g(x)=\sqrt{x}

(principal square root). Which statement is correct? (a) Both

f\bigl(g(x)\bigr)=x

for all

x\ge 0

and

g\bigl(f(x)\bigr)=x

for all

x\in\mathbb R

(b)

g\bigl(f(x)\bigr)=x

for every

x\in\mathbb R

, but

f\bigl(g(x)\bigr)\neq x

for some

x\ge 0

(c)

f\bigl(g(x)\bigr)=x

for every

x\ge 0

, but

g\bigl(f(x)\bigr)\neq x

for some

x\in\mathbb R

(d) Neither composition equals

x

for all admissible

x