Problem 65 - Precalculus

Question

In Exercises \(57-66,\) state the domain and range of the functions. $$y=-2+\frac{1}{2} \sec \left(\pi x+\frac{\pi}{2}\right)$$

Step-by-Step Solution

Answer

Domain: All real numbers except \(x = \frac{2n+1}{2}\); Range: \((-, -2.5] \cup [-1.5, +)\).

Step 1: Identify Function Type

The given function is of the form involving the secant function, which is a trigonometric function. Specifically, it is of the form \(y = a + b \cdot \sec(c \cdot x + d)\). Here, \(a = -2\), \(b = \frac{1}{2}\), \(c = \pi\), and \(d = \frac{\pi}{2}\).

Step 2: Determine Undefined Values for Secant

The secant function, \(\sec(\theta)\), is undefined wherever \(\cos(\theta) = 0\). The angle \(\theta\) in our function is \(\pi x + \frac{\pi}{2}\). Thus, set \(\pi x + \frac{\pi}{2} = (2n+1)\frac{\pi}{2}\) for integers \(n\). Solving for \(x\) gives \(x = \frac{2n+1}{2}\). These are the values where the function is undefined.

Step 3: Define the Domain

The domain of the function is all real numbers except for the values of \(x\) found in Step 2, where the secant is undefined. Therefore, the domain is \(x \in \mathbb{R}\), \(x eq \frac{2n+1}{2}\) for any integer \(n\).

Step 4: Determine the Range of the Function

The range of a general secant function \(y = a + b \cdot \sec(\theta)\) is determined by \(b\). For \(b = \frac{1}{2}\), the function's range extends beyond its vertical asymptotes at the undefined points. Since the secant function has values outside the interval \([-1, 1]\), here the adjusted range is determined by \[-2 \pm \frac{1}{2}\]. Thus, the range is \((-, -2.5] \cup [-1.5, +)\).

Step 5: Conclude with Domain and Range

Summarize the domain and range based on the calculations. The domain is the set of all real numbers excluding \(x = \frac{2n+1}{2}\), and the range is \((-, -2.5] \cup [-1.5, +)\).

Key Concepts

Understanding Domain and Range

In mathematics, the domain and range are fundamental concepts that describe the input and output values a function can have. The domain of a function includes all possible input values \(x\) for which the function is defined. For trigonometric functions like the secant, we must consider where the function could become undefined. A function becomes undefined at inputs where it would need to divide by zero or take some undefined mathematical action.

For the given secant function, replacements that force the expression inside the cosine to equal zero are excluded from the domain.

• Domain: The set of all real numbers except where the cosine value of the expression equals zero because \(\sec(\theta)\) is undefined.
• For our example, this means excluding \(x = \frac{2n+1}{2}\) for any integer \(n\).

The range of a function, on the other hand, consists of all possible output values \(y\) the function can produce. For the secant function, the range is usually all real numbers except those values between \(-1\) and \(1\), translated by any vertical shifts, here indicated by the constants in our formula.

• Range: Everything outside the interval \(-1, 1\), adjusted for any transformations present in the function.

Exploring the Secant Function

The secant function \(\sec(\theta)\) is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the cosine function, \(\sec(\theta) = \frac{1}{\cos(\theta)}\).

Some crucial characteristics of the secant function include:

• The secant function is undefined wherever \(\cos(\theta) = 0\).
• It has vertical asymptotes at intervals of \(\frac{\pi}{2}\), where cosine is zero.
• It is periodic with a period of \(2\pi\), which means its pattern repeats every \(2\pi\) units.

In this exercise, these characteristics are crucial for understanding where the domain is restricted and how the function behaves graphically. Moreover, because any transformations will affect its range, it's pivotal to understand the nature of the base secant function before applying transformations.

Effect of Function Transformation

Function transformations modify the visual output of a function graph, either through shifts, stretches, or compressions. In the function \(y = a + b \cdot \sec(c \cdot x + d)\), each constant \(a\), \(b\), \(c\), and \(d\) affects the graph in different ways:

• \(a\) represents a vertical shift. It moves the entire graph of the function up or down.
• \(b\) is a vertical stretch or compression factor, making the graph stretch away from the x-axis or compress toward it. If \(b\) is negative, it also reflects the function across the x-axis.
• \(c\) affects the frequency of the function. A larger \(c\) value compresses the graph horizontally, increasing the frequency, whereas a smaller \(c\) value stretches it, decreasing the frequency.
• \(d\) represents a horizontal shift. It moves the entire graph to the left or right.

These transformations help to understand how the function \(y = -2 + \frac{1}{2} \sec(\pi x + \frac{\pi}{2})\) will look compared to the basic form of the secant function. Adjusting each element allows us to tailor the function to a specific graph pattern based on desired characteristics.