Finding the Domain of a Function Step by Step
What is the domain of a function? In what is a function we saw that we can think of a function as a rule. This rule, given an input, gives an output.
You could also say that it is a rule that "maps" the input to the output. In calculus, we often use this notation:
This means that "y" is a function of "x". Here, x is the input, often called the independent variable. And y is the output, and it is called the dependent variable.
So, back to the domain of a function. What is it? It is just the set of values that x can take. You can think of the domain as a bag. This bag contains all the x's you can choose as input for the function.
The domain of a function can be defined explicitly or implicitly, but it is always defined.
As an example of a domain defined explicitly, let's say I give you the expression:
Here I tell you that x must be greater than 0. You can't choose any x. In the bag you only have positive x's.
On the other hand, if I simply tell you that:
This function has an implicitly defined domain. I don't specify the valid values of x. So, it is implicit that the domain is the set of all real numbers.
A more interesting example of an implicitly defined domain is the function:
At first glance you may think this is the same as the previous case. However, what would happen if x=2? We'll get
And 1/0 doesn't make sense. (if you're not completely sure why division by zero doesn't make sense, here's a simple explanation).
Because f(2) doesn't make sense, we take the 2 out of the bag, and the domain is the set of all real numbers that are not 2.
Let's consider the function:
In the real numbers, the squareroot of a number is defined only for positive numbers. The squareroots of negative numbers do exist, but we won't consider them here.
So, our function is only defined when there is a positive number inside (or zero!) the square root sign. That means that x-3 must be positive:
Adding 3 to both sides:
And the domain of this function is the set of all numbers equal or greater than 3. A fancy way of saying this is that the domain is:
This is an interval that includes 3 and all numbers greater than 3.
- The domain of a function is the set of all values the independent variable can take.
- The domain can be specified explicitly or implicitly. When it is implicit, the domain is the set of all real numbers for which the function makes sense.
Return from Domain of a Function to Mathematical Functions
Return to Home Page