Browse Questions

Given a function $f:R_* \rightarrow R_*$ defined by $f(x)=\frac{1}{x}$, where $R_*$ is the set of all non-zero real numbers. Which of the following is true?

A) $f$ is one-one only

B) $f$ is onto only

C) $f$ is many-one and onto

D) $f$ is one-one and onto

Toolbox:
• A function $f: A \rightarrow B$ where for every $x1, x2 \in X, f(x1) = f(x2) \Rightarrow x1 = x2$ is called a one-one function.
• A function$f : X \rightarrow Y$ is said to be onto or surjective, if every element of Y is the image of some element of X under f, i.e., for every $y \in Y$, there exists an element x in X such that $f(x) = y$.
Given $f:R_* \rightarrow R_*$ defined by $f(x)=\Large \frac{1}{x}$ where $R_*$ is a set of nonzero real numbers:
Let $x$ and $y$ be two elements in $R_*$. For a one-one function, $f(x) = f(y)$
$\Rightarrow \Large \frac{1}{x}=\frac{1}{y}$$\Rightarrow x = y. Therefore f:R_* \rightarrow R_* defined by f(x)=\Large \frac{1}{x} is one-one. For an on-to function, for every y \in Y, there exists an element x in X such that f(x) = y. \Rightarrow For every y \in R_* there must exist x=\large \frac{1}{y}$$ \in R_*$ such that $f(x)=\frac{1}{\Large ( \frac{1}{y})}$$=y$.
Therefore $f:R_* \rightarrow R_*$ defined by $f(x)=\Large \frac{1}{x}$ is onto.
Solution: $f:R_* \rightarrow R_*$ defined by $f(x)=\Large \frac{1}{x}$ is one-one and onto.