Which Statement Describes The Domain Of The Function $f(x)=\frac{3 X}{4 X^2-4}$?A. All Real Numbers B. All Nonzero Real Numbers C. All Real Numbers Except $x=\frac{3}{4}$ D. All Real Numbers Except $x=-1$ And $x=1$

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Introduction

When dealing with rational functions, it's essential to understand the concept of the domain. The domain of a function is the set of all possible input values (x-values) for which the function is defined. In the case of a rational function, the domain is restricted by the values of x that make the denominator equal to zero. In this article, we will explore the domain of the function f(x)=3x4x2−4f(x)=\frac{3 x}{4 x^2-4} and determine which statement describes it.

Understanding the Function

The given function is a rational function, which means it is the ratio of two polynomials. The numerator is 3x3x, and the denominator is 4x2−44x^2-4. To find the domain of this function, we need to identify the values of x that make the denominator equal to zero.

Finding the Values that Make the Denominator Equal to Zero

To find the values that make the denominator equal to zero, we need to solve the equation 4x2−4=04x^2-4=0. This is a quadratic equation, and we can solve it by factoring or using the quadratic formula.

Factoring the Quadratic Equation

We can factor the quadratic equation 4x2−4=04x^2-4=0 as follows:

4x2−4=04x^2-4=0

4(x2−1)=04(x^2-1)=0

x2−1=0x^2-1=0

(x−1)(x+1)=0(x-1)(x+1)=0

This tells us that either x−1=0x-1=0 or x+1=0x+1=0. Solving for x, we get:

x−1=0⇒x=1x-1=0 \Rightarrow x=1

x+1=0⇒x=−1x+1=0 \Rightarrow x=-1

Therefore, the values that make the denominator equal to zero are x=1x=1 and x=−1x=-1.

Determining the Domain

Since the denominator cannot be equal to zero, we need to exclude the values x=1x=1 and x=−1x=-1 from the domain. This means that the domain of the function f(x)=3x4x2−4f(x)=\frac{3 x}{4 x^2-4} is all real numbers except x=1x=1 and x=−1x=-1.

Conclusion

In conclusion, the domain of the function f(x)=3x4x2−4f(x)=\frac{3 x}{4 x^2-4} is all real numbers except x=1x=1 and x=−1x=-1. This is because the denominator cannot be equal to zero, and the values x=1x=1 and x=−1x=-1 make the denominator equal to zero.

Answer

The correct answer is:

D. All real numbers except x=−1x=-1 and x=1x=1

Final Thoughts

Understanding the domain of a function is crucial in mathematics, especially when dealing with rational functions. By identifying the values that make the denominator equal to zero, we can determine the domain of the function and ensure that we are working with a well-defined function. In this article, we explored the domain of the function f(x)=3x4x2−4f(x)=\frac{3 x}{4 x^2-4} and determined that the correct answer is D. All real numbers except x=−1x=-1 and x=1x=1.

Introduction

In our previous article, we explored the domain of the function f(x)=3x4x2−4f(x)=\frac{3 x}{4 x^2-4} and determined that the correct answer is D. All real numbers except x=−1x=-1 and x=1x=1. In this article, we will answer some frequently asked questions about the domain of a rational function.

Q: What is the domain of a rational function?

A: The domain of a rational function is the set of all possible input values (x-values) for which the function is defined. In other words, it is the set of all real numbers for which the function is not undefined.

Q: How do I find the domain of a rational function?

A: To find the domain of a rational function, you need to identify the values of x that make the denominator equal to zero. You can do this by solving the equation a(x−h)2+k=0a(x-h)^2+k=0, where a(x−h)2+ka(x-h)^2+k is the denominator of the rational function.

Q: What happens if the denominator of a rational function is equal to zero?

A: If the denominator of a rational function is equal to zero, the function is undefined at that point. This means that the value of x that makes the denominator equal to zero is not in the domain of the function.

Q: Can a rational function have a domain of all real numbers?

A: Yes, a rational function can have a domain of all real numbers if the denominator is never equal to zero. This means that the rational function has no restrictions on its domain.

Q: Can a rational function have a domain of all nonzero real numbers?

A: Yes, a rational function can have a domain of all nonzero real numbers if the denominator is never equal to zero and the numerator is never equal to zero. This means that the rational function has no restrictions on its domain, except that x cannot be equal to zero.

Q: How do I determine the domain of a rational function with a quadratic denominator?

A: To determine the domain of a rational function with a quadratic denominator, you need to factor the denominator and identify the values of x that make it equal to zero. You can then exclude these values from the domain.

Q: Can a rational function have a domain of all real numbers except for a single value?

A: Yes, a rational function can have a domain of all real numbers except for a single value if the denominator is never equal to zero, except at that single value. This means that the rational function has no restrictions on its domain, except that x cannot be equal to that single value.

Q: How do I determine the domain of a rational function with a polynomial denominator?

A: To determine the domain of a rational function with a polynomial denominator, you need to identify the values of x that make the denominator equal to zero. You can do this by factoring the denominator and identifying the values of x that make it equal to zero. You can then exclude these values from the domain.

Conclusion

In conclusion, the domain of a rational function is the set of all possible input values (x-values) for which the function is defined. To find the domain of a rational function, you need to identify the values of x that make the denominator equal to zero and exclude these values from the domain. By following these steps, you can determine the domain of a rational function and ensure that you are working with a well-defined function.

Final Thoughts

Understanding the domain of a rational function is crucial in mathematics, especially when dealing with rational functions. By identifying the values that make the denominator equal to zero, we can determine the domain of the function and ensure that we are working with a well-defined function. In this article, we answered some frequently asked questions about the domain of a rational function and provided some tips and tricks for determining the domain of a rational function.