The domain of the tangent function is all real numbers except odd multiples of .

The function is .

The function is undefined when .

The domain of the tangent function is all real numbers except odd multiples of .

Therefore, the domain of is . 

The range of the tangent function is all real numbers.

The function is .

Range depends on the function.

Normally the tangent function would have range from to .

For any values of , .

Therefore, the range of the tangent function is all real numbers.

The tangent function is an odd function, its graph is symmetric with respect to the origin.

The function is .

Trigonometric even-odd identity : .

Since , the tangent function is an odd function.

Since the tangent function is an odd function, its graph is symmetric with respect to the origin.

Period of the tangent function is .

The function is .

If be the point on the unit circle that corresponds to , then is the point on the unit circle that corresponds to .

Thus, .

Suppose that there exists a number , , for which for all .

Then, if , then .

But this means that is a multiple of .

Since no multiple of exists in the interval , this is a contradiction.

Therefore, the period of is .

Vertical asymptotes : , where k is an integer.

Definition of vertical asymptotes :

Vertical asymptotes are straight lines of the equation, toward which a function approaches infinitesimally closely, but never reaches the line, as increases without bound.For these values of , the function is either unbounded or is undefined.

Since the tangent function is undefined at , the vertical asymptotes are .

The domain of the tangent function is all real numbers except odd multiples of .

The range of the tangent function is all real numbers.

The tangent function is an odd function, its graph is symmetric with respect to the origin.

Period of the tangent function is .

Vertical asymptotes : .