Question

\(g(x)=\frac{1}{3} \tan 2 \pi x\)

Short answer

The domain of the function \( g(x) = \frac{1}{3} \tan 2 \pi x \) is \(\mathbb{R}\) except at \( x = \frac{1 + 2k}{4} \), for \( k \) an integer, where it is undefined. The range of this function is \((-\infty, \infty)\) except at these values.

Step-by-step solution

Identify the Domain of the Function

Firstly, we identify the domain of function \( g(x) \). Since the denominator can't be zero, the values of \( x \) where the function \( \tan 2 \pi x \) are undefined correspond to the values where \( 2 \pi x = \frac{\pi}{2} + k\pi \) for \( k \) an integer.

Find the Values of \( x \)

Next, we solve the equality \( 2 \pi x = \frac{\pi}{2} + k\pi \) for \( x \). We find that \( x = \frac{1 + 2k}{4} \), for \( k \) an integer.

Calculate the Range of the Function

Then, we note that the range of the function \( \frac{1}{3} \tan 2 \pi x \) is \((-\infty, \infty)\) except at \( x = \frac{1 + 2k}{4} \), for \( k \) an integer, where it is undefined.

Key concepts

Domain of a Function

Understanding the domain of a function is key to knowing where the function is defined. The domain includes all the possible input values (usually represented by \(x\)) for which the function gives a real result. For the function \(g(x)=\frac{1}{3} \tan 2 \pi x\), the domain excludes the values that make the tangent undefined. The tangent function is undefined whenever its argument equals \(\frac{\pi}{2} + k\pi\), where \(k\) is any integer.

This arises because at these points, the tangent function approaches infinity and thus can’t produce a real result. To find these excluded \(x\) values for \(\tan 2 \pi x\), solve the equation \(2\pi x = \frac{\pi}{2} + k\pi\). Solving gives \(x = \frac{1 + 2k}{4}\) for integer \(k\). Thus, the domain of \(g(x)\) is all real numbers except \(x = \frac{1 + 2k}{4}\).

• Check the values that make the function undefined when identifying the domain.
• The domain includes all input values for which the function operation is valid.

Range of a Function

The range of a function is the set of all possible output values the function can produce. For \(g(x) = \frac{1}{3} \tan 2 \pi x\), the range is significantly defined by the behavior of the tangent function. The tangent function is known for having outputs that span all real numbers as its input angles approach certain undefined points.

However, because \(\tan 2\pi x \) is undefined at points where \(x = \frac{1 + 2k}{4}\) (and \(k\) is an integer), those points are out of range for this function. As a result, \(g(x)\) can take any real value except at these specific \(x\) values where the function becomes undefined.

• The range includes all output values except at the undefined input points.
• Understanding the core trigonometric function helps predict the range behavior.

Undefined Values

In the context of functions, particularly trigonometric ones, undefined values are those where the function does not produce a real number result. For \(g(x)=\frac{1}{3} \tan 2 \pi x\), these undefined values happen when the \(\tan 2 \pi x\) part of the function reaches infinity. This occurs at \(x = \frac{1 + 2k}{4}\), where \(k\) is an integer, because at these points, the tangent function aligns with its vertical asymptotes.

Recognizing undefined values is essential, as it impacts both the domain and range of the function. It helps to systematically identify points where calculations might lead to divisions by zero or infinite results.

• Vertical asymptotes mark the points where the tangent function is undefined.
• Prevent division by zero for functions involving fractions to understand where the function will be undefined.