# Domain and Range of Rational Function

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Posted by Mark M on June 17, 2002 at 19:43:03:

In Reply to: Range & Domain question posted by Bill on June 17, 2002 at 15:00:47:

Hi Bill:

Please remember to put parentheses around numerators or denominators when the meaning is not clear.

I'm going to assume that the fraction in f(x) is NOT just sqrt(x)/x^2. You can type f(x) like this:

f(x) = sqrt(x)/(x^2 + x - 2)

For the domain and range, you came up with:

Domain = {0}

Range = {all real numbers}

I see that, for determining the domain, you factored the denominator, and then excluded the values of -2 and 1. But I don't understand how you went from there to exclude all the rest of the real numbers except zero.

The domain is not zero.

In addition to being concerned about division by zero, we also need to be concerned about taking the square root of a negative number.

Square roots are not defined for negative numbers in the real number system.

Since the numerator of the function is sqrt(x), we know right away that x must be zero or more. So, this tells us that the domain is x >= 0, so far. NOW we can exclude the number(s) that make the denominator zero.

-2 is not >= 0, so -2 has already been excluded from the domain. That leaves just 1 to be excluded.

Domain = {all real numbers >= 0 except 1}

If we write this in interval notation, the domain is:

(0,1) U (1,oo)

Did you ever look at the graph of this rational function? The range may be easier for you to understand if you realize that the behavior of a rational function around its vertical asymptotes
is generally toward positive or negative infinity.

Let us know if you need more help understanding the domain and range of a rational function.

~ Mark

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