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.
Example 1
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.
Example 2
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.
More examples...
Example 3
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.
Conclusion
- 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.
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