Find the Domain and Range of this Function

Question 9: Find the Domain and Range of a Function,$$ f(x) = \tan x $$

Solution:

Given that the function being, $ f(x) = \tan x $ within the interval $ -2\pi \leq x \leq 2\pi $.

1. Understanding the Tangent Function:
The tangent function is defined as: \[ \tan x = \frac{\sin x}{\cos x}. \]
Therefore, $ \tan x $ is undefined wherever $ \cos x = 0 $ because division by zero is not possible.

2. Finding the Domain:
Within the interval $ -2\pi \leq x \leq 2\pi $, the cosine function equals zero at: \[ x = \frac{\pi}{2} + k \pi \quad \text{where } k \in \mathbb{Z}, \] which represents the points where tangent is undefined.
For the interval $ -2\pi \leq x \leq 2\pi $, these points are:

$ x = -\frac{3\pi}{2} $
$ x = -\frac{\pi}{2} $
$ x = \frac{\pi}{2} $
$ x = \frac{3\pi}{2} $

Thus, the domain of $ f(x) = \tan x $ is all real numbers in the given interval, excluding these points:

=$ \{ x \mid -2\pi \leq x \leq 2\pi, x \neq -\frac{3\pi}{2}, -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2} \} $

3. Finding the Range:
Since the base of the exponential function $ b = 2 $ is positive and greater than 1, the function will grow exponentially as $ x $ increases.
The coefficient $ a = 3 $ scales the output, and the constant $ c = -1 $ shifts the entire function downward by 1.
Since exponential functions with positive bases never produce zero or negative results, the minimum value of $ 3 \cdot 2^x $ is 0. This result is shifted down by 1, meaning the minimum output is: \[ \text{Minimum Value} = -1 \] As $ x $ increases, $ f(x) $ continues to grow towards positive infinity.
Therefore, the range of the function is: \[ \text{Range} = (-1, \infty) \]