Finding the Domains of Rational Expressions
Purplemath
What is a rational expression?
A "rational expression" is a polynomial fraction; with variables at least in the denominator. (If variables are only in the numerator, then the expression is actually only linear or a polynomial.) Pretty much anything you could do with regular fractions you can do with rational expressions.
However, since there are variables in rational expressions, there are some additional considerations.
When you dealt with fractions, you knew that the fraction could have any whole numbers for the numerator and denominator, as long as you didn't try putting zero as the denominator.
When dealing with rational expressions, you will often need to evaluate the expression, and it can be useful to know which values would cause division by zero, so you can avoid these x-values. So probably the first thing that they'll have you do with rational expressions is find their domains.
How do you find the domain of a rational expression?
To find the domain of a rational function:
- Take the denominator of the expression.
- Set that denominator equal to zero.
- Solve the resulting equation for the zeroes of the denominator.
- The domain is all other x-values.
- Find the domain of
The domain is all values that x is allowed to be. I can't divide by zerp — because division by zero is never allowed. So I need to find all values of x that would cause division by zero. The domain will then be all other x-values.
When is this denominator equal to zero? When x = 0.
By definition of rational expressions, the domain is the opposite of the solutions to the denominator. When you set the denominator equal to zero and solve, the domain will be all the other values of x. In this case, that means that the domain is:
all x ≠ 0
- Determine the domain of
The domain doesn't care what is in the numerator of a rational expression. The domain is only influenced by the zeroes of the denominator. And that denominator is 3.
Will 3 ever equal zero? No; of course not. And since the denominator will never equal zero, no matter what the value of x is, then there are no forbidden values for this expression, and x can be anything. So the domain is:
all x
Note: In this case, what they gave us was really just a linear expression. It wasn't actually rational, because there were no variables in the denominator.
- Find the domain of the following expression:
To find the domain, I'll ignore the "x + 2" in the numerator (since the numerator does not cause division by zero) and instead I'll look at the denominator. I'll set the denominator equal to zero, and solve. The x-values in the solution will be the x-values which would cause division by zero. The domain will then be all other x-values.
x2 + 2x − 15 = 0
(x + 5)(x − 3) = 0
x = −5, x = 3
By factoring the quadratic, I found the zeroes of the denominator. The domain will then be all other x-values:
all x ≠ −5, 3
- Find the domain of the following expression:
To find the domain, I'll solve for the zeroes of the denominator:
x2 + 4 = 0
x2 = −4
This equation has no solution, so the denominator is never zero. Then the domain is:
all x
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