Find The Domain Of The Rational Expression:$ F(x) = \frac{3}{x^2 + 2x - 35} }$Choose The Answer That Represents The Domain A. { (-∞, -5) \cup (-5, 7) \cup (7, ∞)$ $B. { (-∞, 3) \cup (3, ∞)$} C . \[ C. \[ C . \[ (-∞, -7) \cup (-7,

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Understanding the Concept of Domain

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 expressions, the domain is affected by the values of the variable that make the denominator equal to zero. In this article, we will explore how to find the domain of a rational expression and apply this concept to the given function.

What is a Rational Expression?

A rational expression is a fraction in which the numerator and denominator are polynomials. Rational expressions can be simplified, added, subtracted, multiplied, and divided, just like regular fractions. However, when dealing with rational expressions, we must be careful not to divide by zero, as this would result in an undefined value.

The Given Function

The given function is:

f(x)=3x2+2x−35{ f(x) = \frac{3}{x^2 + 2x - 35} }

To find the domain of this function, we need to determine the values of x that make the denominator equal to zero.

Factoring the Denominator

The denominator of the function is a quadratic expression that can be factored as follows:

x2+2x−35=(x+7)(x−5){ x^2 + 2x - 35 = (x + 7)(x - 5) }

Now that we have factored the denominator, we can set each factor equal to zero and solve for x.

Solving for x

Setting each factor equal to zero, we get:

x+7=0orx−5=0{ x + 7 = 0 \quad \text{or} \quad x - 5 = 0 }

Solving for x, we get:

x=−7orx=5{ x = -7 \quad \text{or} \quad x = 5 }

These are the values of x that make the denominator equal to zero.

Determining the Domain

Since the denominator cannot be equal to zero, we must exclude the values x = -7 and x = 5 from the domain. Therefore, the domain of the function is all real numbers except -7 and 5.

Choosing the Correct Answer

Now that we have determined the domain of the function, we can choose the correct answer from the options provided.

A. (−∞,−5)∪(−5,7)∪(7,∞){(-∞, -5) \cup (-5, 7) \cup (7, ∞)} B. (−∞,3)∪(3,∞){(-∞, 3) \cup (3, ∞)} C. (−∞,−7)∪(−7,5)∪(5,∞){(-∞, -7) \cup (-7, 5) \cup (5, ∞)}

The correct answer is C. (−∞,−7)∪(−7,5)∪(5,∞){(-∞, -7) \cup (-7, 5) \cup (5, ∞)}

Conclusion

In conclusion, finding the domain of a rational expression involves determining the values of the variable that make the denominator equal to zero. By factoring the denominator and solving for x, we can determine the values that must be excluded from the domain. In this article, we applied this concept to the given function and chose the correct answer from the options provided.

Final Answer

The final answer is C. (−∞,−7)∪(−7,5)∪(5,∞){(-∞, -7) \cup (-7, 5) \cup (5, ∞)}

Additional Tips and Tricks

  • When dealing with rational expressions, always check the denominator for zero.
  • Factoring the denominator can help simplify the expression and make it easier to determine the domain.
  • Solving for x by setting each factor equal to zero can help determine the values that must be excluded from the domain.

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about finding the domain of a rational expression.

Q: What is the domain of a rational expression?

A: The domain of a rational expression 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 expression?

A: To find the domain of a rational expression, you need to determine the values of the variable that make the denominator equal to zero. You can do this by factoring the denominator and solving for x.

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. This means that the value of x that makes the denominator equal to zero must be excluded from the domain.

Q: Can I simplify a rational expression before finding its domain?

A: Yes, you can simplify a rational expression before finding its domain. In fact, simplifying the expression can make it easier to determine the domain.

Q: How do I know if a rational expression is undefined?

A: A rational expression is undefined if the denominator is equal to zero. You can check this by setting the denominator equal to zero and solving for x.

Q: Can I have multiple values that make the denominator equal to zero?

A: Yes, you can have multiple values that make the denominator equal to zero. In this case, you need to exclude all of these values from the domain.

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

A: To write the domain of a rational expression in interval notation, you need to exclude the values that make the denominator equal to zero. You can do this by using the union symbol (∪) to combine the intervals.

Q: Can I have a rational expression with no domain?

A: Yes, you can have a rational expression with no domain. This occurs when the denominator is always equal to zero, regardless of the value of x.

Q: How do I determine if a rational expression has no domain?

A: To determine if a rational expression has no domain, you need to check if the denominator is always equal to zero. You can do this by factoring the denominator and checking if it has any real roots.

Q: Can I have a rational expression with a domain that is all real numbers?

A: Yes, you can have a rational expression with a domain that is all real numbers. This occurs when the denominator is never equal to zero, regardless of the value of x.

Q: How do I determine if a rational expression has a domain that is all real numbers?

A: To determine if a rational expression has a domain that is all real numbers, you need to check if the denominator is never equal to zero. You can do this by factoring the denominator and checking if it has any real roots.

Conclusion

In conclusion, finding the domain of a rational expression involves determining the values of the variable that make the denominator equal to zero. By following the steps outlined in this article, you can become more confident in your ability to find the domain of a rational expression and apply this concept to a variety of mathematical problems.

Additional Tips and Tricks

  • Always check the denominator for zero before finding the domain of a rational expression.
  • Factoring the denominator can help simplify the expression and make it easier to determine the domain.
  • Solving for x by setting each factor equal to zero can help determine the values that must be excluded from the domain.
  • Using the union symbol (∪) can help write the domain of a rational expression in interval notation.
  • Checking if the denominator is always equal to zero can help determine if a rational expression has no domain.
  • Checking if the denominator is never equal to zero can help determine if a rational expression has a domain that is all real numbers.