Problem 57 - Precalculus

Question

In Exercises \(57-66,\) state the domain and range of the functions. $$y=\tan \left(\pi x-\frac{\pi}{2}\right)$$

Step-by-Step Solution

Answer

Domain: all real numbers except \(x = k + 1\); Range: all real numbers.

Step 1: Understanding the function

The function given is a transformation of the basic tangent function. The tangent function, \(y = \tan(x)\), has a period of \(\pi\) and vertical asymptotes every \(n\pi + \frac{\pi}{2}\), where \(n\) is an integer.

Step 2: Identify the transformation

The function can be rewritten as \(y = \tan(\pi(x - \frac{1}{2}))\). Here, the variable \(x\) is shifted and scaled by \(\pi\). The phase shift is \(\frac{1}{2}\), meaning the function is shifted \(\frac{1}{2}\) units horizontally.

Step 3: Determine the domain

The domain of the tangent function excludes points where it is undefined, specifically at the vertical asymptotes. For this function, asymptotes occur at \(\pi x - \frac{\pi}{2} = \frac{\pi}{2} + k\pi\), leading to \(x = k + 1\), where \(k\) is an integer. Therefore, the domain is the set of all real numbers except \(x = k + 1\).

Step 4: Determine the range

The range of the tangent function is all real numbers. Despite the horizontal shift and scaling, the range remains unchanged because the transformation does not affect the values that \(y\) can take.

Key Concepts

Tangent Function

The tangent function, denoted as \(y = \tan(x)\), is one of the fundamental trigonometric functions. It is the ratio of the sine and cosine functions: \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). Because of this ratio, the tangent function becomes undefined wherever the cosine of \(x\) equals zero. Here's a quick rundown of its properties:

• Periodicity: The tangent function is periodic with a period of \(\pi\). This means it repeats its pattern every \(\pi\) units along the x-axis.
• Vertical Asymptotes: These occur at \(x = n\pi + \frac{\pi}{2}\), making the function undefined at these points. Here, \(n\) is an integer.

The graph of \(\tan(x)\) oscillates between positive and negative infinity, crossing the x-axis at integer multiples of \(\pi\), where the function is zero. Understanding these aspects helps in visualizing and analyzing its transformations, which is crucial for tackling more complex trigonometric problems.

Domain and Range

The domain and range of trigonometric functions describe the input values (domain) and possible outcomes (range) of the function.
The domain of the tangent function consists of all real numbers except where it has vertical asymptotes. For a transformed tangent function, such as \(y = \tan(\pi x - \frac{\pi}{2})\), the process of determining its domain involves finding these asymptotes:

• The equation for asymptotes \( \pi x - \frac{\pi}{2} = \frac{\pi}{2} + k\pi \) leads to points \(x = k + 1\).
• As a result, the domain is all real numbers except \( x = k + 1 \), where \( k \) is an integer.

For the range, the tangent function doesn’t face similar exclusions as its domain. Its range remains all real numbers \((-\infty, \infty)\), unaffected by transformations involving level shifts or patterns of repetition. This uniform range highlights its open-ended nature, radically different from other trigonometric functions like sine or cosine, which have bounded ranges.

Function Transformation

Function transformations involve operations that shift, stretch, or reflect the graph of a function across the coordinate plane. For the function \(y = \tan(\pi(x - \frac{1}{2}))\), transformation is evident:

• Horizontal Shift: The transformation here shifts the tangent function \(\frac{1}{2}\) units to the right. This is because \(x\) is replaced with \(x - \frac{1}{2}\), shifting the entire graph horizontally.
• Vertical Stretch/Compression: The multiplicative factor of \(\pi\) scales the period of the tangent function. Instead of repeating every \(\pi\), it is further scaled to repeat at periods that align with the transformations of its asymptotes.

Such transformations are critical when analyzing trigonometric functions. They help us tailor functions to meet specific criteria in modeling real-world phenomena. Recognizing these shifts can enhance skills in graph interpretation and further understanding of more intricate trigonometric relationships.