Consider the function defined by

MATHEMATICS · Multi-part · CBSE · Mathematics · Relations and Functions · Relations and Functions

Problem

Consider the function

p:(0,\infty)\to\mathbb{R}

defined by

p(x)=\dfrac{1}{x}

. Respond to the following. (i) Prove that the range of

p

(0,\infty)

. (ii) For an arbitrary positive real number

k

, solve

p(x)=k

for

x

and discuss the uniqueness of the solution within the given domain. (iii) Taking the codomain of

p

to be

\mathbb{R}^{+}

, determine whether

p

is onto and whether it is one-one. Justify your answer fully.