1. Finding domain and range
Module 2 Lesson 4 Remediation

2. Domain and Range
The domain of a relation is the set of all possible input values.
The input variable is called the independent variable.
The range of a relation is the set of all possible output values.
The output variable is called the dependent variable.
Most often, the domain and range of a graph is
described by intervals rather than lists.

3. Domain of a function.
The domain of a function is all real x’s where the function exists.
If the function contains a denominator, set the denominator ≠ 0 and solve for x.
If the function contains a square root, set the argument of the root ≥ 0 and solve for x.
If the function has no square roots and no denominators, the domain is (-∞, ∞).

4. Ex. Find domain and range of the relations shown below.
Domain: [-3, 1] or
Range: [-1, 3] or
Range:

5. Ex. Find domain and range of the relations below

6. Solutions Domain: All Real Numbers or (-infinity, +infinity)
Domain: x ≥ 1 or [1, +infinity)
Range: y ≥ 0 or [0, +infinity)
Range: y ≤ 0 or (-infinity, 0]

7. Domain of a function.
Find the domain of:

8. Domain of a function - Solutions.
Find the domain of:
Solve: 1- 2x ≥ 0
x ≤ 1/ Domain is (-infinity, ½)]
Solve: x – 5 ≠ 0
x ≠ Domain is (-infinity, 5) U (5, +infinity)

9. Example: Find the domain and range of the following graph:
What's the domain?
The graph above is represented by y = x2, and we can square any number we want.
Therefore, the domain is all real numbers.
On a graph the domain corresponds to the horizontal axis.  Since that is the case, we need to look to the left and right to see if there are any end points or holes in the graph to help us find our domain. If the graph keeps going on and on to the right and the graph keeps going on and on to the left then the domain is represented by all real numbers.

10. What's the range?
If I plug any number into this function, am I ever going to be able to get a negative number out of it? No, not in the Real Number System!
The range of this function is all positive numbers which is represented by y ≥ 0.
On a graph, the range corresponds to the vertical axis.  We need to look up and down to see if there are any end points or holes to help us find our range. If the graph keeps going up and down with no endpoint then the range is all real numbers. However, this is not the case.
The graph does not begin to touch the y-axis until x = 0, then it continues up with no endpoints which is represented by y ≥ 0.

11. Let's try another example:
What numbers can we plug into this function?
What happens if we plug in 4?
If x = 4, we divide by zero which is undefined.
Therefore, the domain of this function is: all real numbers except 4.
The range is all real numbers except 0.
(We can only produce zero when a zero is in the numerator.)
In general, when you're trying to find the domain of a function, there are two things you should look out for:
Look for potential division by zero.
Look for places where you might take the square root of a negative number.
If you have a verbal model, you can only use numbers that
make sense in the given situation.