Algebra Help
Domain and Range
When we look at a function like f(x) = x + 2, we can easily say where the function f(x), in this case, the value along the y-axis, is at any point we choose along the x-axis. We know that at x = 3.4, f(x) = 5.4. The values of x that the function will output a value for is called the domain of f. In the case of f(x) = x + 2, the domain is the entire x-axis, or, formally speaking, all real numbers.
The values which f(x) takes on is called the range of f. Since you can always find an x to give you any value for f(x) (i.e., if we want f(x) = 401, then x = 399), the range of f here is the entire y-axis, which we can again say is all real numbers.
If we are looking at a graph, the domain is the x-axis values that the function exists on and the range is the y-axis that the function takes on.
If we have a simple function like a polynomial, the domain and range are often all real numbers. However, domain and range become complicated when we introduce a function that has restrictions on values. The function f(x) =
has a limited domain because we cannot take the square root of negative numbers. Then the domain is all positive real numbers, as well as zero. This set is called the set of non-negative numbers. Since the square root of a number is also positive, the range of f is also the set of non-negative numbers.
Another problem with polynomials arises because of the restriction that w
e cannot divide by zero.
Example 1: Consider the following function:
The function f does not have a value when the denominator, (x-1)(x+8), is equal to zero. Therefore the domain does not include the values x = 1 and x = -8. Note that the graph of the function, shown on the right, reflects the problems at x = 1 and x = -8. In the case of this function, the domain is all real numbers except 1 and -8, and the range is all real numbers.
Exercises
List the Domain and the Range for the following functions. Remember to check for zeroes in the denominator and negatives underneath the square root.
1.
Domain:_______________________ Range:___________________
2.
Domain:_______________________ Range:___________________
3.
Domain:_______________________ Range:___________________
4.
Domain:_______________________ Range:___________________
