Problem 6 - Beginning and Intermediate Algebra

Question

Find the domain of each rational expression. See Example 1. $$ f(x)=\frac{-4 x}{-2+x} $$

Step-by-Step Solution

Answer

The domain of \( f(x) \) is all real numbers except \( x = 2 \).

Step 1: Identify the Rational Expression

The given rational expression is \( f(x) = \frac{-4x}{-2+x} \). A rational expression is a fraction with a polynomial in the numerator and another polynomial in the denominator.

Step 2: Recognize the Domain of a Rational Expression

The domain of a rational expression includes all real numbers except those that make the denominator zero, as division by zero is undefined.

Step 3: Set the Denominator Equal to Zero

To find values that are not in the domain, set the denominator equal to zero: \(-2 + x = 0\). Solve this equation to find the value that must be excluded from the domain.

Step 4: Solve the Equation

Solve the equation \(-2 + x = 0\) by adding 2 to both sides: \[ x = 2 \] This means whenever \(x = 2\), the denominator becomes zero, which is not allowed.

Step 5: State the Domain Exclusion and Conclusion

Since the denominator becomes zero at \(x = 2\), \(x = 2\) is not included in the domain of the function. The domain of \( f(x) \) is all real numbers except \( x = 2 \).

Key Concepts

Domain of a Function

When we talk about the domain of a function, we're discussing all the possible input values (often represented by \(x\)) that a function can accept without causing mathematical issues. For rational expressions, like the one in the exercise, a primary concern is dealing with the denominator. This is because division by zero is undefined in mathematics. Hence, to find the domain of a rational function, you must determine the values of \(x\) that will cause the denominator to be zero. Let's explore how to do that:

• Identify the expression in the denominator: First, we need to look at the equation in the denominator.
• Set the denominator equal to zero: To find problematic values of \(x\), solve the equation by setting the denominator equal to zero.
• Solve for \(x\): This will give you the values of \(x\) that cannot be included in the domain.

After identifying these "excluded" values, you can define the domain, which includes all real numbers except those specific values. This is an essential skill, especially in courses involving algebra and calculus.

Polynomials

Polynomials are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. They are the building blocks of rational expressions, as both the numerator and the denominator of rational expressions are typically polynomials. Here's a quick breakdown of what a polynomial can look like:

• Terms: Individual parts of the polynomial, separated by + or - signs, such as \(-4x\) in the nominator or \(-2 + x\) in the denominator.
• Coefficients: The numbers in front of the terms, for example, -4 in \(-4x\).
• Variables: Symbols like \(x\) that can take various values within the domain.

Polynomials are versatile. They help in modeling various phenomena in science and engineering. Understanding their structure simplifies the process of manipulating them, especially when tackling rational expressions. By examining polynomials closely, you can gain insights into the overall behavior of the rational functions they compose.

Fractional Equations

A fractional equation involves fractions containing polynomials. Solving them often requires finding a common denominator or setting up an equation to isolate one variable. In the context of rational expressions, the primary concern is ensuring the denominator does not equate to zero. Here's how to effectively approach fractional equations:

• Clear the denominator: You might need to multiply through by the least common denominator to eliminate fractions, though in domain questions, the aim is simply understanding the values making the denominator zero.
• Balance the equation: This involves algebraic manipulation to keep the equation valid and solve it accurately.
• Check solutions: Especially important in fractional equations, as sometimes solving can yield an extraneous solution—one that doesn't satisfy the original equation.

By adhering to these methods, mastering fractional equations becomes straightforward, allowing you to better understand broader mathematical concepts. Recognizing where issues might arise due to zero denominators helps solidify your grip on rational expressions and their behaviors.