Answer:
We are given the tangent function \(f(x)=\tan x\).
Firstly we know that, \(\tan x=\frac{\sin x}{\cos x}\), where \(\sin x\) is the sine function and \(\cos x\) is the cosine function.
Now, tangent function will be zero when its numerator is zero.
i.e. \(\tan x=0\) when \(\sin x=0\).
i.e. \(\tan x=0\) when \(x=n \pi\), where n is the set of integers.
So, tangent function crosses x-axis at \(n \pi\), n is the set of integers.
Further, tangent function will be undefined when its denominator is zero.
i.e. \(\tan x=0\) when \(\cos x=0\).
i.e. \(\tan x=0\) when \(x=(2n-1) \frac{\pi}{2}\), where n is the set of integers.
Moreover, a zero in the denominator gives vertical asymptotes.
So, tangent function will have vertical asymptotes at \((2n-1) \frac{\pi}{2}\), n is the set of integers.
Therefore, these key features gives us the graph of a tangent function as shown below.
