Find The Domain Of The Function: F ( X ) = X 2 + X X 2 − 7 X + 10 F(x)=\frac{x^2+x}{x^2-7x+10} F ( X ) = X 2 − 7 X + 10 X 2 + X A) All Real Numbers Except 2 And 5 B) All Real Numbers Except -2 And -5 C) All Real Numbers Except 2 And -5 D) All Real Numbers Except -2 And 5

Mar 8, 2025 by ADMIN 277 views

**Finding the Domain of a Rational Function**

Introduction

In mathematics, the domain of a function is the set of all possible input values for which the function is defined. When dealing with rational functions, the domain is restricted by the values of x that make the denominator equal to zero. In this article, we will explore how to find the domain of the rational function $f(x)=\frac{x^2+x}{x^2-7x+10}$ .

Understanding Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the given function $f(x)=\frac{x^2+x}{x^2-7x+10}$ , the numerator is a quadratic polynomial, and the denominator is also a quadratic polynomial. The domain of a rational function is restricted by the values of x that make the denominator equal to zero.

Finding the Domain

To find the domain of the function $f(x)=\frac{x^2+x}{x^2-7x+10}$ , we need to find the values of x that make the denominator equal to zero. We can do this by solving the equation $x^2-7x+10=0$ .

Step 1: Factor the Denominator

The first step in solving the equation $x^2-7x+10=0$ is to factor the denominator. We can factor the quadratic expression as $(x-2)(x-5)=0$ .

Step 2: Solve for x

Now that we have factored the denominator, we can solve for x by setting each factor equal to zero. We get two equations: $x-2=0$ and $x-5=0$ . Solving these equations, we get $x=2$ and $x=5$ .

Step 3: Exclude the Values of x

Since the denominator cannot be equal to zero, we need to exclude the values of x that make the denominator equal to zero. In this case, we need to exclude $x=2$ and $x=5$ from the domain.

Conclusion

In conclusion, the domain of the function $f(x)=\frac{x^2+x}{x^2-7x+10}$ is all real numbers except $x=2$ and $x=5$ . This means that the function is defined for all real numbers except $x=2$ and $x=5$ .

Answer

The correct answer is A) All real numbers except 2 and 5.

Final Thoughts

Introduction

In our previous article, we explored how to find the domain of a rational function by identifying the values of x that make the denominator equal to zero. In this article, we will answer some frequently asked questions about finding 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 for which the function is defined. In other words, it is the set of all real numbers except those that make the denominator equal to zero.

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

A: To find the domain of a rational function, you need to follow these steps:

Factor the denominator of the rational function.
Set each factor equal to zero and solve for x.
Exclude the values of x that make the denominator equal to zero from the domain.

Q: What if the denominator is not factorable?

A: If the denominator is not factorable, you can use other methods such as the quadratic formula to find the values of x that make the denominator equal to zero.

Q: Can the domain of a rational function be empty?

A: Yes, the domain of a rational function can be empty if the denominator is always equal to zero. This means that the function is not defined for any real number.

Q: How do I determine if a rational function is defined at a particular point?

A: To determine if a rational function is defined at a particular point, you need to check if the denominator is equal to zero at that point. If the denominator is not equal to zero, then the function is defined at that point.

Q: Can the domain of a rational function be a single point?

A: Yes, the domain of a rational function can be a single point if the denominator is equal to zero at that point, but the numerator is not equal to zero.

Q: How do I graph a rational function?

A: To graph a rational function, you need to follow these steps:

Find the domain of the function.
Plot the points in the domain on a coordinate plane.
Draw a smooth curve through the points to represent the function.

Q: Can a rational function have a domain that is a union of intervals?

A: Yes, a rational function can have a domain that is a union of intervals. For example, the domain of the function $f(x)=\frac{x^2-4}{x-2}$ is the union of the intervals $(-\infty, 2)$ and $(2, \infty)$ .

Conclusion

In conclusion, finding the domain of a rational function is an important step in understanding the behavior of the function. By following the steps outlined in this article, you can determine the domain of a rational function and ensure that you are working with a well-defined function.

Final Thoughts

Finding the domain of a rational function is a critical step in understanding the behavior of the function. By identifying the values of x that make the denominator equal to zero, you can determine the domain of the function and ensure that you are working with a well-defined function. In this article, we have seen how to find the domain of a rational function and answer some frequently asked questions about the topic.