Determine Which Values Of X Are Not In The Domain Of The Rational Function G(x) = 4x / (x^2 - 3x - 10).

In mathematics, especially when dealing with functions, understanding the domain is crucial. The domain of a function is the set of all possible input values (often denoted as x) for which the function will produce a valid output. However, certain functions, like rational functions, have restrictions on their domains. These restrictions arise because certain input values can lead to undefined operations, such as division by zero. This article delves into how to identify these excluded values, focusing on the rational function:

$$g(x) = \frac{4x}{x^2 - 3x - 10}$$

We will explore the steps necessary to determine the values of x that are not in the domain of this function, ensuring a comprehensive understanding of the concept.

Understanding Rational Functions and Their Domains

To effectively find the values excluded from the domain, we must first understand what rational functions are and why they have domain restrictions. Rational functions are functions that can be expressed as the quotient of two polynomials. In simpler terms, they are fractions where the numerator and the denominator are both polynomials. Our example function, $g(x) = \frac{4x}{x^2 - 3x - 10}$, perfectly fits this definition. The numerator, 4x, is a polynomial, and the denominator, x² - 3x - 10, is also a polynomial.

The primary reason rational functions have domain restrictions lies in the denominator. Division by zero is undefined in mathematics. Therefore, any value of x that makes the denominator of a rational function equal to zero must be excluded from the domain. These excluded values are the roots, or zeroes, of the denominator polynomial. To find these values, we set the denominator equal to zero and solve for x. This process will help us pinpoint the specific x values that would lead to an undefined result, ensuring we accurately define the function's domain.

Finding the Excluded Values: Setting the Denominator to Zero

To determine the values of x that are not in the domain of the given rational function, the crucial step is to identify the values that make the denominator equal to zero. The function we are working with is:

$$g(x) = \frac{4x}{x^2 - 3x - 10}$$

The denominator is the quadratic expression x² - 3x - 10. As we've established, we need to find the values of x for which this expression equals zero. This is because division by zero is undefined, and any such x values must be excluded from the domain of g(x). Mathematically, we express this by setting the denominator equal to zero:

$$x^2 - 3x - 10 = 0$$

This equation represents a quadratic equation, which can be solved using various methods, such as factoring, completing the square, or using the quadratic formula. In this case, factoring is a straightforward approach. By finding the roots of this quadratic equation, we will identify the x values that make the denominator zero, and thus, are not part of the domain of the rational function. The next step will involve factoring this quadratic equation to find these critical values.

Factoring the Quadratic Expression

Having set the denominator equal to zero, our next task is to solve the quadratic equation:

$$x^2 - 3x - 10 = 0$$

The most efficient method for solving this particular quadratic equation is factoring. Factoring involves expressing the quadratic expression as a product of two binomials. We are looking for two numbers that multiply to -10 (the constant term) and add up to -3 (the coefficient of the x term). These two numbers are -5 and 2, since (-5) * (2) = -10 and (-5) + 2 = -3. Therefore, we can rewrite the quadratic equation in factored form as:

$$(x - 5)(x + 2) = 0$$

This factored form allows us to easily identify the values of x that make the expression equal to zero. According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. This principle is fundamental in solving factored equations. In our case, it means that either (x - 5) must equal zero, or (x + 2) must equal zero. This leads us to two separate linear equations that we can solve for x. The solutions to these equations will be the values that are excluded from the domain of the original rational function.

Solving for x: Identifying the Excluded Values

From the factored form of our quadratic equation, (x - 5)(x + 2) = 0, we can now determine the values of x that make the equation true. We apply the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This gives us two separate equations to solve:

1. x - 5 = 0
2. x + 2 = 0

Solving the first equation, x - 5 = 0, we add 5 to both sides to isolate x:

$$x = 5$$

This tells us that when x is 5, the factor (x - 5) becomes zero, thus making the entire expression zero.

Next, we solve the second equation, x + 2 = 0. We subtract 2 from both sides to isolate x:

$$x = -2$$

This indicates that when x is -2, the factor (x + 2) is zero, again making the entire expression zero.

Therefore, the values of x that make the denominator of our rational function zero are 5 and -2. These are the values that are not in the domain of the function. The domain of the function includes all real numbers except these two values.

Defining the Domain of the Rational Function

Having identified the values of x that make the denominator of our rational function zero, we are now ready to define the domain of the function. We found that x = 5 and x = -2 are the values that must be excluded. This is because these values would result in division by zero, which is undefined in mathematics. Therefore, the domain of the function g(x) includes all real numbers except 5 and -2.

We can express the domain in several ways. One common way is to use set notation. In set notation, the domain of g(x) can be written as:

$$\{x \in \mathbb{R} \mid x \neq 5, x \neq -2 \}$$

This notation reads as "the set of all x in the set of real numbers such that x is not equal to 5 and x is not equal to -2." Another way to express the domain is using interval notation. In interval notation, we represent the domain as a union of intervals:

$$(-\infty, -2) \cup (-2, 5) \cup (5, \infty)$$

This notation indicates that the domain includes all real numbers from negative infinity up to -2, not including -2, then all numbers between -2 and 5, not including -2 and 5, and finally, all numbers from 5 to positive infinity, not including 5. Both set notation and interval notation provide precise ways to describe the domain of a rational function, clearly indicating the values that are permissible inputs.

Conclusion: Key Takeaways for Domain Determination

In this article, we have thoroughly explored the process of finding the values that are not in the domain of a rational function. Our focus was on the function:

$$g(x) = \frac{4x}{x^2 - 3x - 10}$$

We began by understanding that the domain of a function consists of all possible input values for which the function produces a valid output. For rational functions, the key restriction is that the denominator cannot be zero. To find the excluded values, we set the denominator equal to zero and solved for x. In this case, the denominator was the quadratic expression x² - 3x - 10, which we factored into (x - 5)(x + 2). Setting each factor to zero, we found that x = 5 and x = -2 are the values that make the denominator zero.

Therefore, these values are not in the domain of g(x). We then expressed the domain using both set notation and interval notation, demonstrating different ways to clearly communicate the allowable input values for the function. The domain, in interval notation, is:

$$(-\infty, -2) \cup (-2, 5) \cup (5, \infty)$$

Understanding how to determine the domain of a rational function is a fundamental skill in mathematics. By identifying and excluding values that lead to undefined operations, we ensure that we are working within the valid scope of the function. This process not only provides a complete understanding of the function's behavior but also lays the groundwork for more advanced mathematical concepts and applications.