Problem 5 - A Graphical Approach to College Algebra

Question

Find the domain of each rational expression. $$\frac{-8}{x^{2}+1}$$

Step-by-Step Solution

Answer

The domain is all real numbers, since \(x^2 + 1 \neq 0\) for any real \(x\).

Step 1: Understanding the Domain of a Rational Expression

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

Step 2: Identify the Denominator

Look at the rational expression, \(-\frac{8}{x^2 + 1}\). Here, the denominator is \(x^2 + 1\).

Step 3: Determine the Values that Make the Denominator Zero

To find any restrictions on the domain, set the denominator equal to zero and solve: \(x^2+1=0\).

Step 4: Solve the Equation for Real Values

Subtract 1 from both sides to obtain \(x^2 = -1\). Since \(x^2\) cannot be negative for any real number, this equation has no real solutions.

Step 5: Conclusion on Domain

Since there are no real numbers that make the denominator zero, the rational expression is defined for all real values of \(x\). Therefore, the domain is all real numbers.

Key Concepts

Domain of a Function

Determining the domain of a function is an essential aspect of understanding how the function behaves and what inputs it can accept. To find the domain, we need to identify the set of all possible input values, often represented by the variable \(x\), for which the function produces a valid output. For rational expressions, the domain is influenced by the denominator, as it cannot be zero.

Generally, the domain of a function excludes values that result in mathematical impossibilities, such as division by zero. In our specific case, the function is \(-\frac{8}{x^2 + 1}\). The domain will be all real numbers because the expression in the denominator, \(x^2 + 1\), will never equal zero for any real \(x\).

To summarize:

• The domain is the set of acceptable values for \(x\).
• For rational expressions, ensure the denominator is not zero.
• For \(-\frac{8}{x^2 + 1}\), the domain is all real numbers because \(x^2 + 1\) is always positive for real \(x\).

Denominator

The denominator is the part of a fraction that appears below the fraction line, and in a division operation, it represents the divisor. In rational expressions, the denominator plays a critical role, specifically in defining the function's domain.

In our example \(-\frac{8}{x^2 + 1}\), the denominator is \(x^2 + 1\).The characteristics of the denominator, such as never being able to become zero for real numbers, ensure that our rational expression is defined across all real numbers. This happens because:

• \(x^2\) is non-negative for all real \(x\).
• Adding 1 to \(x^2\) results in a positive denominator for all real \(x\).

Thus, the denominator not only affects the domain but also the behavior of the function since it ensures that the expression is free from undefined points due to zero division.

Division by Zero

Division by zero is a mathematical impossibility, as it doesn't produce a finite or useful result. When a denominator of a fraction equals zero, the expression becomes undefined.

In the realm of rational expressions like \(-\frac{8}{x^2 + 1}\), avoiding division by zero is crucial. Setting the denominator to zero and solving for \(x\) helps identify any values excluded from the domain. However, since \(x^2 + 1\) never equals zero (due to \(x^2\) being always non-negative), there are no such values in this case.

Remember:

• Division by zero results in an undefined expression.
• Always check the denominator for values of \(x\) that might cause zero division.
• For \(-\frac{8}{x^2 + 1}\), division by zero is never a concern for real \(x\).

Keeping these points in mind ensures we avoid undefined scenarios when working with rational expressions.