Question

Define an identity function and draw its graph, also find its domain and range.

Answer 1

Let $R$ be the set of real numbers. Thus, the real-valued function $f:R\to R$ by $y=f\left(a\right)=a$ for all $a\in R$, is called the identity function. Here the domain and range (co-domain) of function $f$ are $R$. Hence, each element of set $R$ has an image on itself. The graph is a straight line and it passes through the origin. The application of this function can be seen in the identity matrix. Mathematically it can be expressed as;

$f\left(a\right)=a\forall a\in R$

Identity Function Graph

If we plot a graph for identity function, then it will appear to be a straight line. Let us plot a graph for function say $f\left(x\right)=x$, by putting different values of $x$.

$x$	$-2$	$-1$	$0$	$1$	$2$
$f\left(x\right)=y$	$-2$	$-1$	$0$	$1$	$2$

Now as you can see from the above table, the values are the same for both $x-axis$ and $y-axis$. Hence, let us plot a graph based on these values.

Hence, from the above graph, it is clear that the identity function gives a straight line in the $xy-plane$.

And it is defined from the set of real numbers. Therefore, we have the domain and the range as the set of all real numbers, that is,
Domain of the function$=R$
Range of the function $=R$