Question

If $A\left\{ { - 2, - 1,1,2} \right\}$ and $f = \left\{ {\left( {x,\dfrac{1}{x}} \right):x \in A} \right\}$, write down the range of $f$. Is $f$a function from $A$ to $A$?

Hint: First of all, find the range of the given function by substituting the values of $A$ in the given function. The function $f$a function from $A$ to $A$ exists if both its domain and range are equal. So, use this concept to reach the solution of the given problem.

Given $A = \left\{ { - 2, - 1,1,2} \right\}$
And $f = \left\{ {\left( {x,\dfrac{1}{x}} \right):x \in A} \right\}$
$\Rightarrow f\left( { - 2} \right) = \dfrac{1}{{ - 2}} = \dfrac{{ - 1}}{2} \\ \Rightarrow f\left( { - 1} \right) = \dfrac{1}{{ - 1}} = - 1 \\ \Rightarrow f\left( 1 \right) = \dfrac{1}{1} = 1 \\ \Rightarrow f\left( 2 \right) = \dfrac{1}{2} \\$
So, the range of the function $f$ is $\left\{ {\dfrac{{ - 1}}{2}, - 1,1,\dfrac{1}{2}} \right\}$.
If we consider the function $f$ from $A$ to $A$ i.e., $f:A \to A$ here it is clear that the domain and range of the function $f$ are the same.
But clearly from the above data we have different values for domain and range of the function $f$.
Hence the function $f$ from $A$ to $A$ i.e., $f:A \to A$ can`t exist.
Thus, $f$ from $A$ to $A$ is not a function.
Note: Function is a relation in which each element of the domain is paired with exactly one element of the range. Range of a function is also known as the co-domain of the function. The function $f$ from $A$ to $A$ can be related as $f:A \to A$.