Which Equation Represents A Tangent Function With A Domain Of All Real Numbers Such That $x \neq \frac{\pi}{2} + \pi N$, Where $n$ Is An Integer?A. F ( X ) = Tan ⁡ ( 2 X − Π F(x) = \tan (2x - \pi F ( X ) = Tan ( 2 X − Π ] B. G ( X ) = Tan ⁡ ( X − Π G(x) = \tan (x - \pi G ( X ) = Tan ( X − Π ] C.

The tangent function is a fundamental concept in trigonometry, and it plays a crucial role in various mathematical applications. In this article, we will explore the tangent function and its domain, and we will determine which equation represents a tangent function with a domain of all real numbers such that $x \neq \frac{\pi}{2} + \pi n$, where $n$ is an integer.

What is the Tangent Function?

The tangent function is a trigonometric function that is defined as the ratio of the sine and cosine functions. It is denoted by the symbol $\tan(x)$ and is defined as:

$$\tan(x) = \frac{\sin(x)}{\cos(x)}$$

The tangent function has a periodic nature, meaning that it repeats itself at regular intervals. The period of the tangent function is $\pi$, which means that the function repeats itself every $\pi$ units.

Domain of the Tangent Function

The domain of a function is the set of all possible input values for which the function is defined. In the case of the tangent function, the domain is all real numbers except for the values that make the denominator zero. Since the denominator is $\cos(x)$, the tangent function is undefined when $\cos(x) = 0$.

The cosine function is zero at odd multiples of $\frac{\pi}{2}$, which means that the tangent function is undefined at these values. Therefore, the domain of the tangent function is all real numbers except for the values of the form $\frac{\pi}{2} + \pi n$, where $n$ is an integer.

Which Equation Represents a Tangent Function with a Domain of All Real Numbers?

Now that we have understood the tangent function and its domain, we can examine the given equations and determine which one represents a tangent function with a domain of all real numbers such that $x \neq \frac{\pi}{2} + \pi n$, where $n$ is an integer.

Equation A: $f(x) = \tan (2x - \pi)$

This equation represents a tangent function with a period of $\frac{\pi}{2}$, which is half the period of the standard tangent function. The graph of this function will have a period of $\frac{\pi}{2}$, and it will be undefined at values of the form $\frac{\pi}{4} + \pi n$, where $n$ is an integer.

Equation B: $g(x) = \tan (x - \pi)$

This equation represents a tangent function with a period of $\pi$, which is the same as the period of the standard tangent function. The graph of this function will have a period of $\pi$, and it will be undefined at values of the form $\frac{\pi}{2} + \pi n$, where $n$ is an integer.

Equation C: $h(x) = \tan (x)$

This equation represents the standard tangent function, which has a period of $\pi$ and is undefined at values of the form $\frac{\pi}{2} + \pi n$, where $n$ is an integer.

Conclusion

Based on our analysis, we can conclude that Equation B, $g(x) = \tan (x - \pi)$, represents a tangent function with a domain of all real numbers such that $x \neq \frac{\pi}{2} + \pi n$, where $n$ is an integer.

The graph of this function will have a period of $\pi$, and it will be undefined at values of the form $\frac{\pi}{2} + \pi n$, where $n$ is an integer. This equation meets the given conditions, and it represents a tangent function with the desired domain.

Final Answer

In this article, we will answer some frequently asked questions about the tangent function and its domain.

Q: What is the tangent function?

A: The tangent function is a trigonometric function that is defined as the ratio of the sine and cosine functions. It is denoted by the symbol $\tan(x)$ and is defined as:

$$\tan(x) = \frac{\sin(x)}{\cos(x)}$$

Q: What is the domain of the tangent function?

A: The domain of the tangent function is all real numbers except for the values that make the denominator zero. Since the denominator is $\cos(x)$, the tangent function is undefined when $\cos(x) = 0$.

The cosine function is zero at odd multiples of $\frac{\pi}{2}$, which means that the tangent function is undefined at these values. Therefore, the domain of the tangent function is all real numbers except for the values of the form $\frac{\pi}{2} + \pi n$, where $n$ is an integer.

Q: What is the period of the tangent function?

A: The period of the tangent function is $\pi$, which means that the function repeats itself every $\pi$ units.

Q: Which equation represents a tangent function with a domain of all real numbers such that $x \neq \frac{\pi}{2} + \pi n$, where $n$ is an integer?

A: Equation B, $g(x) = \tan (x - \pi)$, represents a tangent function with a domain of all real numbers such that $x \neq \frac{\pi}{2} + \pi n$, where $n$ is an integer.

Q: What is the graph of the tangent function?

A: The graph of the tangent function is a periodic function that has a period of $\pi$. The graph has vertical asymptotes at values of the form $\frac{\pi}{2} + \pi n$, where $n$ is an integer.

Q: How do I graph the tangent function?

A: To graph the tangent function, you can use a graphing calculator or a computer algebra system. You can also use a table of values to plot the function.

Q: What are some common applications of the tangent function?

A: The tangent function has many applications in mathematics and science. Some common applications include:

• Trigonometry: The tangent function is used to solve trigonometric equations and to find the values of trigonometric functions.
• Calculus: The tangent function is used to find the derivative of a function and to solve optimization problems.
• Physics: The tangent function is used to describe the motion of objects and to solve problems involving circular motion.
• Engineering: The tangent function is used to design and analyze systems that involve circular motion, such as gears and pulleys.

Q: What are some common mistakes to avoid when working with the tangent function?

A: Some common mistakes to avoid when working with the tangent function include:

• Not checking the domain of the function before graphing or solving it.
• Not using the correct period of the function when graphing or solving it.
• Not using the correct values of the function when solving trigonometric equations.
• Not checking for vertical asymptotes when graphing the function.

Conclusion

In this article, we have answered some frequently asked questions about the tangent function and its domain. We have also discussed some common applications of the tangent function and some common mistakes to avoid when working with it.