Q. 132 - Precalculus Enhanced with Graphing Utilities

Question

Write down five properties of the tangent function. Explain the meaning of each.

Step-by-Step Solution

Answer

1. Its domain is $-\infty <x<+\infty$ , except all odd integer multiples of $\frac{\mathrm{\pi }}{2}$ 

2. Its range is $-\infty <y<+\infty$ 

3. The period of this function is P (180 degree) 

4. Function tangent is an odd function i.e. $f(-x)=-f\left(x\right)$

5. It has vertical asymptotes at odd integer multiples of $\frac{\mathrm{\pi }}{2}$ 

Step 1. Given information

Given function is Tangent i.e.$f\left(x\right)=\tan \theta$

Step 2. Five properties of tangent

1. Its domain is $-\infty <x<+\infty$ , except all odd integer multiples of $\frac{\mathrm{\pi }}{2}$ i.e. function is satisfied for all real values of its independent variable x except for odd integer multiples of $\frac{\mathrm{\pi }}{2}$ as at $x=\frac{\mathrm{\pi }}{2}$,$y=\tan x$  is not defined.

2. Its range is $-\infty <y<+\infty$ i.e. the function has all real values, positive or negative, as its solution.

3. The period of this function is P (180 degree) that means that the value of this function starts repeating after the interval of P. If the value $\theta$ is in the domain of the tangent function then $\pi +\theta$ also lies in the domain. So the period of the tangent function is $\pi$.

4. Function tangent is an odd function i.e. $f(-x)=-f\left(x\right)$. Geometrically, the graph of tangent function has rotational symmetry with respect to the origin, it means that its graph remains unchanged after rotation of 180 degrees about the origin.

5. It has vertical asymptotes at odd integer multiples of $\frac{\mathrm{\pi }}{2}$ i.e. this function is not defined at odd integer multiples of $\frac{\mathrm{\pi }}{2}$, or its graph don't touch the lines parallel to vertical y- axis (also called vertical asymptotes).