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Learn about input and output values in functions, domain and range concepts, limitations on domain, and real number sets. Understand how to determine the domain and range of a function with examples.
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Domain and Range
As we study functions we learn terms like input values and output values.
Input values are the numbers we put into the function. They are the x-values. Output values are the numbers that come out of the function. They are the y-values.
Given the function, we can choose any value we want for x. Suppose we choose 11. We can put 11 into the function by substituting for x.
If we wrote down every number we could put in for x andstill have the function make sense, we would have the set of numbers we call the domain of the function.
The domain is the set that contains all the input values for a function.
In our function is there any number we could not put in for x? No!
Because we could substitute any real number for x, we say the domain of the function is the set of real numbers.
To use the symbols of algebra, we could write the domain as Does that look like a foreign language? Let’s translate:
The curly braces just tell us we have a set of numbers.
The x reminds us that our set contains x-values.
The colon says, such that
The symbol that looks like an e (or a c sticking its tongue out) says, belongs to . . .
And the cursive, or script, R is short for the set of real numbers.
So we read it, “The set x belongs to of x such that R, the set of real numbers.”
When we put 11 in for x, y was 17.
So 17 belongs to the range of the function, Is there any number that we could not get for y by putting some number in for x?
No! We say that the range of the function is the set of real numbers.
Read this: “The set of y, such that y belongs to R, the set of real numbers.”
It is not always true that the domain and range can be any real number. Sometimes mathematicians want to study a function over a limited domain.
They might think about the function where x is between –3 and 3. It could be written,
Sometimes the function itself limits the domain or range. In this function, can x be any real number?
What would happen if x were 3? We can never divide by 0. Then we would have to divide by 0.
So we would have to eliminate 3 from the domain. The domain would be,
Can you think of a number which could not belong to the range? y could never be 0. Why?
What would x have to be for y to be 0? The range of the function is, There is no number we can divide 1 by to get 0, so 0 cannot belong to the range.
Rule 2: You can’t take the square root of a negative number. The most common rules of algebra that limit the domain of functions are: Rule 1: You can’t divide by 0.
We’ve already seen an example of Rule 1: You can’t divide by 0.
Think about Rule 2, You can’t take the square root of a negative number. Given the function, what is the domain?
What is y when x is 16? The square root of 16 is 4, so y is 4 when x is 16 16 belongs to the domain, and 4 belongs to the range.
But what is y when x is –16? What number do you square to get –16? Did you say –4?
not –16. There is no real number we can square to get a negative number. So no negative number can belong to the domain of
The smallest number for which we can find a square root is 0, so the domain of is
