Question

Show that the range of the tangent function is the set of all real numbers

Short answer

As tangent function takes values from $-\infty to\infty$, we can conclude that the range of tangent function is set of all real values.

Step-by-step solution

Step 1. Given Information

We have to show that the range of a tangent function is set of all real numbers.

Step 2. Explanation

We know that a tangent function $y=\tan x$is a periodic function of period $\mathrm{\pi }$. Also it continuous from $-\frac{\mathrm{\pi }}{2}to\frac{\mathrm{\pi }}{2}$.

The value of $\tan \left(\frac{\mathrm{\pi }}{2}\right)=\infty$and the value of $\tan \left(-\frac{\mathrm{\pi }}{2}\right)=-\infty$. Also $\tan \left(0\right)=0$.

From these valuers we can see that the tan functions takes values from $-\infty$to $\infty$.

So we can conclude that the range of tangent function is set of all real values.