2a. Domain and Range of a Function
Definitions of Domain and Range
Domain
The domain of a function is the complete set of possible values of the independent variable in the function.
In plain English, this definition means:
The domain of a function is the set of all possible x values which will make the function "work" and will output real y-values.
When finding the domain, remember:
- The denominator (bottom) of a fraction cannot be zero
- The values under a square root sign must be positive
Example: The function y = √(x + 4) has the following graph.
The domain of the function is x ≥ −4, since x cannot take values less than −4. (Try some values in your calculator, some less than −4 and some more than −4. The only ones that "work" and give us an answer are the ones greater than or equal to −4).
Note:
- The enclosed (colored-in) circle on the point (-4, 0). This indicates that the domain "starts" at this point.
- That x can take any positive value in this example.
Range
The range of a function is the complete set of all possible resulting values of the dependent variable of a function, after we have substituted the values in the domain.
In plain English, the definition means:
The range of a function is the possible y values of a function that result when we substitute all the possible x-values into the function.
When finding the range, remember:
- Substitute different x-values into the expression for y to see what is happening
- Make sure you look for minimum and maximum values of y
- Draw a sketch!
Example 1: Let's return to the example above, y = √(x + 4). We notice that there are only positive y-values. There is no value of x that we can find such that we will get a negative value of y. We say that the range for this function is y ≥ 0.
Example 2: The curve of y = sin x shows the range to be betweeen −1 and 1.
The domain of the function y = sin x is "all values of x", since there are no restrictions on the values for x.
More Domain and Range Examples
You can see more examples of domain and range in the section Inverse Trigonometric Functions.
Note: Because we are assuming that only real numbers are to be used in the domain and range of a function, values that lead to division by zero or to imaginary numbers are not included. The Complex Numbers chapter explains more about imaginary numbers.
Example 1
(a) Find the domain and range for the function
f(x) =
x2 + 2.
(b) Find the domain and range for the function
Example 2
Find the domain and range for the function 
In general, we determine the domain of each function by looking for those values of the independent variable which cannot be used.
The range of each function is found through an
inspection of the function.
Example 3
Find the domain and range for the function defined as
f(x) = x2 + 4 for x > 2
Example 4
More Domain and Range Examples
In case you missed it earlier, you can see more examples of domain and range in the section Inverse Trigonometric Functions.
We are told that the height h, in metres, of a certain projectile as a function of time t, in seconds, is
h = 20t − 4.9t2
Find the domain and range for the function h(t).
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