Problem 63 - Precalculus

Question

In Exercises \(57-66,\) state the domain and range of the functions. $$y=-3 \tan \left(\frac{\pi}{4} x-\pi\right)+1$$

Step-by-Step Solution

Answer

Domain: All real numbers except \(x = 6 + 4k\); Range: \((-\infty, \infty)\).

Step 1: Understanding the Tangent Function

The tangent function, \(y = \tan(x)\), has vertical asymptotes where the function is undefined. These occur at \(x = \frac{\pi}{2} + k\pi\) for integer values of \(k\). The range of \(y = \tan(x)\) is all real numbers.

Step 2: Transforming the Function

The given function is \(y = -3 \tan \left(\frac{\pi}{4} x - \pi\right) + 1\). Here, \(\frac{\pi}{4} x - \pi\) indicates a horizontal transformation. We need to find where \(\frac{\pi}{4} x - \pi = \frac{\pi}{2} + k\pi\) (where the tangent is undefined).

Step 3: Solving for Undefined Points

Set \(\frac{\pi}{4} x - \pi = \frac{\pi}{2} + k\pi\) and solve for \(x\): \[\frac{\pi}{4} x - \pi = \frac{\pi}{2} + k\pi\] \[\frac{\pi}{4} x = \frac{3\pi}{2} + k\pi\] \[x = 6 + 4k\] This means the function is undefined at \(x = 6 + 4k\), where \(k\) is an integer.

Step 4: Determining the Domain

The domain of the function excludes the points where it is undefined: \(x eq 6 + 4k\) for integer values of \(k\). Therefore, the domain is all real numbers except \(x = 6 + 4k\).

Step 5: Determining the Range

The vertical transformations affect the range of the function. The constant \(-3\) causes a vertical stretch by a factor of 3 and a reflection, and adding 1 shifts the graph upwards by 1 unit. However, since the tangent can take any real number, the range remains all real numbers. Thus, the range is \((-\infty, \infty)\).

Key Concepts

Domain and Range

In mathematics, every function has particular characteristics known as domain and range. The domain is essentially the set of all possible input values for a given function. For our tangent function, this is where the function will output a real number. However, the tangent function has vertical asymptotes where it is undefined. These points occur at various intervals depending on the function's transformations.
For the given function, \(y = -3 \tan \left(\frac{\pi}{4} x - \pi\right) + 1\), we determined that the asymptotes happen at \(x = 6 + 4k\), for integer values of \(k\). This means, the domain excludes these points:

• The formula for the undefined points is \(x = 6 + 4k\).
• The domain includes all real numbers except these points.

The range of the tangent function, no matter how it is transformed, remains as all real numbers \((-\infty, \infty)\). Why? Because the transformations only change the shape and position of the curve, not the capability of the function to produce any real number.

Tangent Function

The tangent function is one of the primary trigonometric functions. It is usually expressed as \(y = \tan(x)\). This function is unique among trig functions due to its vertical asymptotes. These asymptotes are points where the tangent function becomes undefined, creating breaks in its graph. For the regular tangent function, these breaks occur at intervals of \(\frac{\pi}{2} + k\pi\) where \(k\) is an integer.
The standard tangent function has a period of \(\pi\), which means it repeats every \(\pi\) units. Thus, any transformation applied to the tangent function affects these intervals.
While the tangent function's range remains all real numbers, the domain must be specifically considered to avoid these undefined points.

Function Transformations

Function transformations involve changing the appearance of a graph through various operations like shifts, stretches, and reflections. For our tangent function, the formula \(y = -3 \tan \left(\frac{\pi}{4} x - \pi\right) + 1\) indicates specific transformations applied:

• Horizontal transformations: Inside the tangent function, \(\frac{\pi}{4} x - \pi\) shifts the graph horizontally. The \(\frac{\pi}{4}\) factor affects the period of the function, stretching it out so the period becomes \(4\pi\).
• Vertical transformations: The \(-3\) in front of the tangent function stretches the graph vertically by a factor of 3. It also reflects the graph over the x-axis. The final \(+1\) shifts the entire graph upwards by one unit.

These transformations do not affect the range of real numbers but do influence where the function is undefined and the appearance of the graph.