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Questions  >>  CBSE XII  >>  Math  >>  Relations and Functions

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?

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  • 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.
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