How to find the domain of rational expressions?

Understand the Problem

The question is asking how to determine the domain of rational expressions, which involves identifying the values for which the expression is defined. Generally, this requires finding points where the denominator is not equal to zero.

Answer

The domain of a rational expression is all real numbers excluding where the denominator equals zero.

Steps to Solve

Identify the rational expression First, identify the rational expression in the problem. A rational expression typically has the form $\frac{N(x)}{D(x)}$, where $N(x)$ is the numerator and $D(x)$ is the denominator.
Set the denominator to zero To find the values that cause the rational expression to be undefined, set the denominator equal to zero: $$ D(x) = 0 $$
Solve for x Solve the equation from the previous step to find values of $x$ that make the denominator zero. These values will be excluded from the domain of the rational expression.
State the domain Express the domain of the rational expression in interval notation by excluding the values found in step 3. If the denominator is zero at $x = a$, the domain is typically written as: $$ (-\infty, a) \cup (a, +\infty) $$

The domain of the rational expression is typically all real numbers except for those values of $x$ where the denominator equals zero.

More Information

Knowing how to determine the domain of rational expressions is crucial in algebra, as it helps ensure that calculations remain valid and avoids division by zero. The domain can sometimes be represented in interval notation, which is a compact way to express all valid input values.

Tips

Forgetting to exclude values where the denominator is zero. Always double-check that the denominator does not equal zero for any values included in the domain.
Not properly expressing the domain in interval notation. Make sure to use parentheses to indicate that the points where the denominator is zero are not included.

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