Find The Domain Of The Rational Function. F ( X ) = 6 X 7 X − 49 F(x)=\frac{6x}{7x-49} F ( X ) = 7 X − 49 6 X ​ The Domain Of The Function Is $\square$ . (Type Your Answer In Interval Notation.)

Introduction

In mathematics, a rational function is a function that can be expressed as the ratio of two polynomials. 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 possible values of x that make the function valid. In this article, we will focus on finding the domain of the rational function $f(x)=\frac{6x}{7x-49}$.

Understanding the Domain of a Rational Function

The domain of a rational function is determined by the values of x that make the denominator of the function equal to zero. This is because division by zero is undefined in mathematics. Therefore, to find the domain of a rational function, we need to find the values of x that make the denominator equal to zero and exclude them from the domain.

Finding the Values of x that Make the Denominator Equal to Zero

To find the values of x that make the denominator equal to zero, we need to set the denominator equal to zero and solve for x. In this case, the denominator is $7x-49$. Setting it equal to zero, we get:

$$7x-49=0$$

Solving for x, we get:

$$7x=49$$

$$x=\frac{49}{7}$$

$$x=7$$

Therefore, the value of x that makes the denominator equal to zero is x=7.

Excluding the Value of x from the Domain

Since the value of x that makes the denominator equal to zero is x=7, we need to exclude this value from the domain. In interval notation, the domain of the function is written as:

$$(-\infty, 7) \cup (7, \infty)$$

This means that the function is defined for all values of x except x=7.

Conclusion

In conclusion, the domain of the rational function $f(x)=\frac{6x}{7x-49}$ is $(-\infty, 7) \cup (7, \infty)$. This means that the function is defined for all values of x except x=7.

Example Problems

1. Find the domain of the rational function $f(x)=\frac{2x}{x+1}$.
2. Find the domain of the rational function $f(x)=\frac{x-2}{x-3}$.
3. Find the domain of the rational function $f(x)=\frac{x+1}{x-2}$.

Solutions

1. The domain of the rational function $f(x)=\frac{2x}{x+1}$ is $(-\infty, -1) \cup (-1, \infty)$.
2. The domain of the rational function $f(x)=\frac{x-2}{x-3}$ is $(-\infty, 3) \cup (3, \infty)$.
3. The domain of the rational function $f(x)=\frac{x+1}{x-2}$ is $(-\infty, 2) \cup (2, \infty)$.

Tips and Tricks

1. When finding the domain of a rational function, always set the denominator equal to zero and solve for x.
2. Exclude the value of x that makes the denominator equal to zero from the domain.
3. Use interval notation to write the domain of the function.

Common Mistakes

1. Failing to set the denominator equal to zero and solve for x.
2. Including the value of x that makes the denominator equal to zero in the domain.
3. Not using interval notation to write the domain of the function.

Real-World Applications

1. Finding the domain of a rational function is important in physics and engineering, where it is used to model real-world phenomena.
2. In economics, the domain of a rational function is used to model the behavior of economic systems.
3. In computer science, the domain of a rational function is used to model the behavior of algorithms and data structures.

Conclusion

Frequently Asked Questions

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.

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

A: To find the domain of a rational function, you need to set the denominator equal to zero and solve for x. Then, exclude the value of x that makes the denominator equal to zero from the domain.

Q: What happens if the denominator is equal to zero?

A: If the denominator is equal to zero, the function is undefined at that point. Therefore, you need to exclude that value from the domain.

Q: Can I include the value of x that makes the denominator equal to zero in the domain?

A: No, you cannot include the value of x that makes the denominator equal to zero in the domain. This is because division by zero is undefined in mathematics.

Q: How do I write the domain of a rational function in interval notation?

A: To write the domain of a rational function in interval notation, you need to use the following format:

$$(a, b)$$

Where a and b are the values that the function is defined for.

Q: What is the difference between the domain and the range of a function?

A: The domain of a function is the set of all possible input values (x-values) for which the function is defined. The range of a function is the set of all possible output values (y-values) that the function can produce.

Q: Can I find the domain of a rational function with a variable in the denominator?

A: Yes, you can find the domain of a rational function with a variable in the denominator. To do this, you need to set the denominator equal to zero and solve for the variable.

Q: How do I find the domain of a rational function with a quadratic expression in the denominator?

A: To find the domain of a rational function with a quadratic expression in the denominator, you need to factor the quadratic expression and set each factor equal to zero. Then, solve for x and exclude the values that make the denominator equal to zero.

Q: Can I find the domain of a rational function with a rational expression in the denominator?

A: Yes, you can find the domain of a rational function with a rational expression in the denominator. To do this, you need to set the denominator equal to zero and solve for x.

Q: What is the importance of finding the domain of a rational function?

A: Finding the domain of a rational function is important because it helps you to determine the set of all possible input values (x-values) for which the function is defined. This is crucial in physics and engineering, where it is used to model real-world phenomena.

Q: Can I use technology to find the domain of a rational function?

A: Yes, you can use technology to find the domain of a rational function. Many graphing calculators and computer algebra systems can help you to find the domain of a rational function.

Q: How do I check my answer for the domain of a rational function?

A: To check your answer for the domain of a rational function, you need to plug in a value from the domain into the function and make sure that the function is defined at that point.

Q: What are some common mistakes to avoid when finding the domain of a rational function?

A: Some common mistakes to avoid when finding the domain of a rational function include:

• Failing to set the denominator equal to zero and solve for x.
• Including the value of x that makes the denominator equal to zero in the domain.
• Not using interval notation to write the domain of the function.

Conclusion

In conclusion, finding the domain of a rational function is an important concept in mathematics. By following the steps outlined in this article, you can find the domain of any rational function. Remember to set the denominator equal to zero and solve for x, exclude the value of x that makes the denominator equal to zero from the domain, and use interval notation to write the domain of the function.