The graph of the function [tex]\( y = \tan(x) \)[/tex] has asymptotes at the values of [tex]\( x \)[/tex] where the function is undefined. Let's go through the detailed reasoning step-by-step:
1. Definition of the Tangent Function:
The tangent function is defined as:
[tex]\[
y = \tan(x) = \frac{\sin(x)}{\cos(x)}
\][/tex]
2. Undefined Points of [tex]\( \tan(x) \)[/tex]:
The tangent function will be undefined wherever the denominator (i.e., [tex]\( \cos(x) \)[/tex]) is equal to zero. This is because division by zero is undefined in mathematics.
3. Condition for Asymptotes:
So, to determine where [tex]\( y = \tan(x) \)[/tex] is undefined, we set the denominator equal to zero:
[tex]\[
\cos(x) = 0
\][/tex]
4. Solutions to [tex]\( \cos(x) = 0 \)[/tex]:
The values of [tex]\( x \)[/tex] where [tex]\( \cos(x) = 0 \)[/tex] can be found within one period of the cosine function, which is [tex]\( 2\pi \)[/tex]. For [tex]\( \cos(x) \)[/tex], these values are:
[tex]\[
x = \frac{\pi}{2} + k\pi \quad \text{for} \quad k \in \mathbb{Z}
\][/tex]
Here, [tex]\( k \)[/tex] is any integer, meaning [tex]\( x \)[/tex] occurs at [tex]\( \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \ldots \)[/tex] and so on.
5. Conclusion:
Therefore, the graph of [tex]\( y = \tan(x) \)[/tex] has vertical asymptotes at the values of [tex]\( x \)[/tex] where [tex]\( \cos(x) = 0 \)[/tex].
The correct answer is:
- at the values of [tex]\( x \)[/tex] where [tex]\( \cos(x) = 0 \)[/tex]
