Domains and Ranges of Rational functions

quotient, rational functions, domain, range, output and input

The domain of a function is the input, or possible inputs, often associated with the x variable.  The range of a function is possible outputs for the different inputs of a function.  It is often associated with the y variable.  The range is often more tricky to figure out than the domain.  However, on a graph, the range merely requires looking at what y-values are achieved by a function.

In Rational functions, domains are easily determined by figuring out where the function is undefined.  Since rational functions are quotients of two polynomials, this merely means that we look at where the denominator of the rational function is undefined.

Ex.  (x²+4x+4)/(x+3) has domain All Real numbers except -3. This is because putting -3 into the function makes the denominator zero, which gives an undefined value.


To find the range of this function, we think of possible y-outputs.  Since the function can achieve all values of y on the real numbers, we see that the range is All Real.

quotient, rational functions, domain, range, output and input