Question

\(g(x)=3 \tan x\)

Short answer

The function \(g(x)=3 \tan x\) is a vertically scaled version of the basic tangent function. The domain of the function is 'all real numbers except \((2n+1)\frac{\pi}{2}\)' where \(n\) is an integer, because the tangent function goes to positive or negative infinity (undefined) at these points. The range is 'all real numbers'. The function is periodic with a period of \(\pi\), and its graph increases between its vertical asymptotes.

Step-by-step solution

Understand the basic tangent function

The basic function \(tan x\) is the quotient of the sine to the cosine function i.e., \(tan x = \frac{sin x}{cos x}\). It's important to recall that the tangent function is periodic with a period of \(\pi\) and each of its periods range from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). Also remember that tangent function is undefined at \(\pm \frac{\pi}{2}\), \( \pm \frac{3\pi}{2}\) and so forth, because cosine becomes zero at these points.

Analyze the given function

The function presented here, \(g(x)=3 \tan x\), is simply the basic tangent function scaled vertically by a factor of 3. It means that the range of the function will remain the same but the maximum and minimum value of the function within each period will be three times to that of \(tan x\), while the period will stay the same.

Define the domain and range of the function

The domain of the function \(g(x)=3 \tan x\) is 'all real numbers except \((2n+1)\frac{\pi}{2}\)' where \(n\) is an integer, because the tangent function has a vertical asymptotes at these points. The range of \(g(x)=3 \tan x\) is 'all real numbers', because the tangent function can take any real value.

Periodicity and general behavior

The function \(g(x)=3 \tan x\) has a period of \(\pi\), which means it repeats its values with an interval of \(\pi\). The graph of this function will have vertical asymptotes at each \(x = (2n+1)\frac{\pi}{2}\), and between these the function is increasing.

Key concepts

Tangent Function

The tangent function, represented as \( \tan x \), is a fundamental trigonometric function defined as the ratio of the sine to the cosine function. This can be expressed mathematically as \( \tan x = \frac{\sin x}{\cos x} \).

The tangent function is unique because it shows how the projection of a point on a unit circle behaves as the angle \( x \) changes. Unlike the sine and cosine functions, which are oscillating, the tangent function has distinct characteristics:

• The function produces undefined values when the cosine of \( x \) equals zero, specifically at odd multiples of \( \frac{\pi}{2} \), such as \( \pm \frac{\pi}{2}, \pm \frac{3\pi}{2}, \) and so forth.
• These undefined points, where the function shoots up to infinity or drops to negative infinity, represent the vertical asymptotes in its graph.
• The tangent function is an odd function, meaning \( \tan(-x) = -\tan(x) \).

This inherent property of producing an undefined value leads to important concepts about its domain, range, and behavior.

Periodicity

Periodicity is a critical aspect of trigonometric functions, including the tangent function. It signifies the repetition of the function values at regular intervals.

For the tangent function, the period is \( \pi \). This means that every \( \pi \) units along the x-axis, the values of \( \tan x \) repeat.

• The function \( g(x) = 3 \tan x \) shares the same periodic behavior with its parent function \( \tan x \).
• This periodicity holds true irrespective of any vertical scaling, such as the factor of 3 in \( g(x) = 3 \tan x \).

As a result, if you were to graph \( g(x) = 3 \tan x \), you would notice that the curve repeats itself over intervals of length \( \pi \), forming identical patterns time and again.

Asymptotes

In terms of trigonometry and function behavior, the asymptotes of a tangent function carry significant importance. They are vertical lines on its graph where the function tends towards positive and negative infinity.

For \( \tan x \), these lines occur at the points where cosine is zero:

• The vertical asymptotes capture the locations \( x = (2n+1)\frac{\pi}{2} \), where \( n \) is any integer. Hence, at \( x = \pm \frac{\pi}{2}, \pm \frac{3\pi}{2}, \ldots \), the function becomes undefined.
• In the case of \( g(x) = 3 \tan x \), the vertical asymptotes remain exactly the same as those of \( \tan x \).

Visually, these asymptotes serve as breaks in the graph, separating each continuous segment of the tangent function's periodic behavior.

Function Domain and Range

Understanding the domain and range of the tangent function is essential for grasping its full behavior.

• **Domain:** For \( g(x) = 3 \tan x \), the domain excludes any values where the function is undefined, specifically multiples of \((2n+1)\frac{\pi}{2} \) due to the vertical asymptotes.
• This is expressed as 'all real numbers except \((2n+1)\frac{\pi}{2} \)', indicating the x-values that are permissible.

• **Range:** Unlike the sine and cosine functions, the range of the tangent function is all real numbers, \((-\infty, \infty) \).
• This attribute remains unchanged even when the function is scaled or transformed. Therefore, \( g(x) = 3 \tan x \) can take any real value in its range.

This broad range allows the tangent function to describe such varied behaviors when plotted, filling up the plane over the uninterrupted domains within each period.