To identify values that make an expression undefined
Identify values that make a rational expression undefined: set the denominator equal to zero and solve to find all x-values that must be excluded from the expression's domain.
Key Concepts
An expression becomes undefined when its denominator equals zero. To find these restricted values, you must inspect the denominators of the original expression, before any simplification. For a division problem $$ \frac{a}{b} \div \frac{c}{d} $$, you must ensure that b, c, and d are all nonzero, as they each pose a risk of division by zero.
Example 1: For $$ \frac{x+1}{(x 5)(x+3)} $$, the expression is undefined when x = 5 or x = 3. Example 2: For $$ \frac{x}{x 6} \div \frac{x+2}{x(x 6)} $$, the expression is undefined for x=6, x= 2, and x=0.
The denominator is like the ground you're standing on; dividing by zero makes it disappear! To find these forbidden values, you must check every single denominator in the original problem before you simplify or cancel anything. Even if a factor like (x 2) cancels out later, it was part of the original setup, making x=2 off limits.
Common Questions
What makes a value undefined in a rational expression?
A value of x is undefined whenever it makes the denominator equal to zero, since division by zero is not allowed. These values must be excluded from the domain of the rational expression.
How do you systematically find all undefined values?
Set the entire denominator polynomial equal to zero. Factor the polynomial if possible, then apply the Zero Product Property to find each root. For (2x)/(x^2-x-6), factor the denominator to (x-3)(x+2)=0, giving excluded values x=3 and x=-2.
Must you check the numerator for undefined values too?
No, undefined values come only from the denominator equaling zero. However, if a factor appears in both numerator and denominator, that value creates a hole rather than a vertical asymptote, though it is still excluded from the domain.