Problem 13 - Algebra for College Students

Question

Find the domain of each rational expression. $$\frac{x-1}{x^{2}+4}$$

Step-by-Step Solution

Answer

The domain is all real numbers.

Step 1 - Identify the rational expression

The given rational expression is \( \frac{x-1}{x^{2}+4} \).

Step 2 - Understand the domain of a rational expression

The domain of a rational expression is all the values of x for which the expression is defined. A rational expression is undefined when its denominator is zero.

Step 3 - Set the denominator not equal to zero

To find the domain, set the denominator \(x^{2}+4\) not equal to zero: \[ x^{2} + 4 eq 0 \]

Step 4 - Solve the inequality

Solve the inequality to find which values of x are allowed. Notice that \(x^{2} + 4 = 0\) leads to \( x^{2} = -4 \). Since squaring a real number never gives a negative result, there is no real solution for this equation.

Step 5 - Determine the domain

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

Key Concepts

Real Numbers

Real numbers are simply all the numbers you know from everyday life. These include:

• Whole numbers like 3, 7, and 10
• Fractions like \(\frac{1}{2}\) and \(\frac{3}{4}\)
• Decimals like 0.5 and 1.75
• Negative numbers like -2 and -5.6

Essentially, a real number can be any value on the number line. Complex numbers, which include imaginary numbers like \(i\) (where \(i\) is the square root of -1), are not counted as real numbers.
In the context of rational expressions, we're mostly interested in real numbers because we want to know which real values make the expression defined or undefined.

Denominator

A denominator is the bottom part of a fraction. For example, in the rational expression \( \frac{a}{b} \), \( b \) is the denominator. In every fraction, the denominator has an important role:

• It tells us how many parts the whole is divided into.
• In rational expressions, the denominator determines which values of \( x \) are allowed.

In our exercise, the given rational expression is \( \frac{x-1}{x^{2}+4} \). Here, \( x^2 + 4 \) is the denominator.
If the denominator is zero, the expression becomes undefined. That’s why we need to check the values of \( x \) that ensure the denominator is not zero.

Undefined Expressions

In mathematics, undefined expressions refer to cases where the expression does not have a clear value. One classic example is division by zero. When the denominator of a fraction is zero, the entire fraction is undefined.
To find if a rational expression becomes undefined, you need to set the denominator equal to zero and solve the equation for \( x \). If there are real solutions to the equation, those particular \( x \) values are excluded from the domain.
For our specific exercise with \( \frac{x-1}{x^2 + 4} \, solving \ ( x^2 + 4 \) not equal to zero shows that there are no real values making the denominator zero. Thus, it's always defined and the domain is all real numbers.