Consider $ y = f(x) = x $ As y = x, both x and y take identical values. That is why this function is called an identity function. Both x and y are defined for all real values of x. So domain and range of the identity function is all real value, that is $ x \epsilon R $ and $ y \epsilon R $.
Graph of the identity function, 'f'(x) = x is a straight line passing through the origin and inclined at an angle of 45o ($ \pi $ with X-axis. It is also known as angle bisector of I and III quadrant.