Help with Domain and Range
Domain: This is the set of all possible values of x. That is, x-values which will produce valid
outputs, or real y-values.
1
, there is a restriction on the x-values. We cannot replace
Example: In the function f ( x ) =
x+2
the x with 2, because it would yield a denominator equal to zero (thus, it would be undefined).
We then say the domain is the set of all real number, with one restriction, x ≠ 2.
Another example: Suppose we have the function
2 , then there is a restriction for x
√
here as well. We can take the square root of a positive number or the square root of 0, but not the
square root of a negative value (it would be undefined); therefore, the domain would be all
values of
2 0, or we can write this as
2. In interval notation, we could write this as
[2, ∞); that is, from 2 to infinity.
Range: This is the set of all output values that can be generated by the function; that is, the
possible y-values that result when we replace all the possible x-values into the function.
Example: If we graph the function
2, we see that the range is all the values greater
than or equal to 2, because when we square any x-value the resulting y-value will always be 0 or
positive, and never negative. Since the smallest y-value " " can give will always be 0 or
2 will give is 2.
positive, the smallest y-value the function
Finding the domain from a graph: Because the domain is related to the x-values, find the
“leftmost or the smallest x-value, then find the “rightmost” or the largest x-value.
Finding the range from a graph: Because the range is related to the y-values, find the “bottom
most” or the smallest y-value, then find the “top most” or the largest y-value.
Example:
6 f(x)=y
5
4
3
2
1
x
-6 -5 -4 -3 -2 -1-1 1 2 3 4 5 6
-2
-3
-4
-5
-6
(Assume graph has endpoints where x = -7 and x = 7)
Here, the Domain goes from x = -7 to x = 7, or simply -7 ≤ x ≤ 7. In interval notation, we write
this as [-7, 7].
The Range goes from y = -5 to y = 5, or simply -5 ≤ y ≤ 5. In interval notation, [-5, 5].
So, remember: Domain refers to the x-values and the Range refers to the y-values.