Find The Domain Of The Function ${ F(x) = \frac{45}{x^2 + 3x - 130} }$

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Introduction

When dealing with rational functions, it's essential to determine the domain of the function, which is the set of all possible input values (x) for which the function is defined. In this article, we will focus on finding the domain of the function f(x) = 45 / (x^2 + 3x - 130). We will use algebraic techniques to identify the values of x that make the denominator of the function equal to zero, as these values will be excluded from the domain.

Understanding the Function

The given function is a rational function, which means it is the ratio of two polynomials. The numerator of the function is a constant, 45, while the denominator is a quadratic expression, x^2 + 3x - 130. To find the domain of the function, we need to determine 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 quadratic equation x^2 + 3x - 130 = 0. We can use the quadratic formula to solve this equation:

x = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 1, b = 3, and c = -130. Plugging these values into the quadratic formula, we get:

x = (-(3) ± √((3)^2 - 4(1)(-130))) / 2(1) x = (-3 ± √(9 + 520)) / 2 x = (-3 ± √529) / 2 x = (-3 ± 23) / 2

Simplifying the expression, we get two possible values for x:

x = (-3 + 23) / 2 = 10 x = (-3 - 23) / 2 = -13

Determining the Domain of the Function

Now that we have found the values that make the denominator equal to zero, we can determine the domain of the function. The domain of the function is all real numbers except for the values that make the denominator equal to zero. In this case, the values that make the denominator equal to zero are x = 10 and x = -13.

Writing the Domain in Interval Notation

To write the domain in interval notation, we need to specify the values that are included and excluded from the domain. In this case, the domain is all real numbers except for x = 10 and x = -13. We can write this in interval notation as:

(-∞, -13) ∪ (-13, 10) ∪ (10, ∞)

Conclusion

In this article, we have found the domain of the function f(x) = 45 / (x^2 + 3x - 130). We used algebraic techniques to identify the values of x that make the denominator equal to zero, and we determined the domain of the function to be all real numbers except for x = 10 and x = -13. We also wrote the domain in interval notation as (-∞, -13) ∪ (-13, 10) ∪ (10, ∞).

Final Answer

The final answer is: (−∞,−13)∪(−13,10)∪(10,∞)\boxed{(-∞, -13) ∪ (-13, 10) ∪ (10, ∞)}

Frequently Asked Questions

Q: What is the domain of the function f(x) = 45 / (x^2 + 3x - 130)?

A: The domain of the function is all real numbers except for x = 10 and x = -13.

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

A: To find the domain of a rational function, you need to determine the values of x that make the denominator equal to zero. You can use algebraic techniques, such as solving quadratic equations, to find these values.

Q: What is the significance of the domain of a function?

A: The domain of a function is the set of all possible input values (x) for which the function is defined. It's essential to determine the domain of a function to ensure that the function is well-defined and can be evaluated for all possible input values.

Q: Can the domain of a function be empty?

A: Yes, the domain of a function can be empty. This occurs when the denominator of the function is always equal to zero, making the function undefined for all possible input values.

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

A: To write the domain of a function in interval notation, you need to specify the values that are included and excluded from the domain. You can use the following notation:

  • (-∞, a) to indicate that all values less than a are included
  • (a, b) to indicate that all values between a and b are included
  • (b, ∞) to indicate that all values greater than b are included
  • ∪ to indicate that the domain is the union of multiple intervals

References

  • [1] "Algebra and Trigonometry" by Michael Sullivan
  • [2] "Calculus" by James Stewart
  • [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Introduction

In our previous article, we discussed how to find the domain of a function, which is the set of all possible input values (x) for which the function is defined. In this article, we will provide a Q&A section to help you better understand the concept of finding the domain of a function.

Q&A

Q: What is the domain of the function f(x) = 1 / (x - 2)?

A: The domain of the function is all real numbers except for x = 2.

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

A: To find the domain of a rational function, you need to determine the values of x that make the denominator equal to zero. You can use algebraic techniques, such as solving quadratic equations, to find these values.

Q: What is the significance of the domain of a function?

A: The domain of a function is the set of all possible input values (x) for which the function is defined. It's essential to determine the domain of a function to ensure that the function is well-defined and can be evaluated for all possible input values.

Q: Can the domain of a function be empty?

A: Yes, the domain of a function can be empty. This occurs when the denominator of the function is always equal to zero, making the function undefined for all possible input values.

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

A: To write the domain of a function in interval notation, you need to specify the values that are included and excluded from the domain. You can use the following notation:

  • (-∞, a) to indicate that all values less than a are included
  • (a, b) to indicate that all values between a and b are included
  • (b, ∞) to indicate that all values greater than b are included
  • ∪ to indicate that the domain is the union of multiple intervals

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) for which the function is defined, while the range of a function is the set of all possible output values (y) that the function can produce.

Q: How do I find the range of a function?

A: To find the range of a function, you need to determine the set of all possible output values (y) that the function can produce. This can be done by analyzing the function and determining the possible values of y.

Q: Can a function have multiple domains?

A: Yes, a function can have multiple domains. This occurs when the function is defined for multiple intervals or sets of values.

Q: How do I determine if a function is continuous or discontinuous?

A: To determine if a function is continuous or discontinuous, you need to analyze the function and determine if it has any gaps or breaks in its graph.

Q: What is the significance of continuity in a function?

A: Continuity in a function is significant because it ensures that the function can be evaluated for all possible input values (x) without any gaps or breaks in its graph.

Conclusion

In this article, we have provided a Q&A section to help you better understand the concept of finding the domain of a function. We have discussed various questions and answers related to the domain of a function, including how to find the domain, the significance of the domain, and how to write the domain in interval notation.

Final Answer

The final answer is: The domain of a function is the set of all possible input values (x) for which the function is defined.

References

  • [1] "Algebra and Trigonometry" by Michael Sullivan
  • [2] "Calculus" by James Stewart
  • [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Additional Resources

  • [1] Khan Academy: Domain and Range of a Function
  • [2] Mathway: Domain and Range of a Function
  • [3] Wolfram Alpha: Domain and Range of a Function