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The tangent function, often denoted as \ \(\tan(x)\ \) , is a fundamental trigonometric function. It is defined as the ratio of the sine and cosine functions: \ \( \tan(x) = \frac{\sin(x)}{\cos(x)} \ \). The tangent function has several unique properties:
The function is periodic, meaning it repeats its values in regular intervals. For the standard tangent function, this period is \ \(\pi\ \).
The tangent function has vertical asymptotes where the cosine function is zero, causing the function to be undefined.
This function can take on any real number value, making its range \ \(( -\infty , \infty )\ \) . In the given problem \ \(y=5 \tan (\pi x+\pi)\ \) , the tangent function is adjusted by coefficients and shifts, but the core properties remain the same, just scaled or shifted accordingly.

The period of a function is the interval after which the function starts repeating its values. For the tangent function \ \(\tan(x)\ \), the period is traditionally \ \(\pi\ \), because \ \(\tan(x + \pi) = \tan(x)\ \).
When the tangent function is modified by multiplying the variable inside the function by a constant \ \(b\ \), the period of the function changes. Specifically, the period becomes \ \(\frac{\pi}{b}\ \).
In the exercise, the function is \ \(y = 5 \tan(\pi x + \pi)\ \). Here, \ \(b = \pi\ \) which simplifies the period to \ \(\frac{\pi}{\pi} = 1\ \).
Therefore, the function repeats its values every 1 unit along the x-axis.

Precalculus is a course designed to prepare students for calculus. It covers important mathematical concepts and functions, including trigonometry, which deals with angles and their relationships. Understanding functions like the tangent function
and its properties is crucial in precalculus as they form the foundation for more advanced topics in calculus. Core concepts you will encounter include:

• Period of a function: Understanding how often a function repeats its values.
• Vertical asymptotes: Knowing where a function becomes undefined or unbounded.
• Range of a function: Learning the set of all possible outputs a function can produce.

Mastering these concepts in precalculus will give you the mathematical tools needed for calculus and other higher-level math courses.