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Let X and Y be set of real number with a rule maps each element x of X to an unique element y of Y then the set X is called the domain of the set. The set Y is called the co domain. 

How do you find the domain and range of a function on a graph?

To find the domain of the function from the graph look from left to right on the graph. See if the graph is a terminating one or it extends infinitely on either side. One should also look for breaks in the graphs and see if at that breaks is y defined or not. An open circle or a vertical asymptote can indicate a break in domain.  When writing down the domain, we may have to exclude the points where there are no y values.
To find the range of the function, look for the lowest y value and highest y value on the curve, if the curve is not extending beyond either way then the range would be from the lowest to the highest.

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Example 1:

Find the domain and range of  the function from the graph. 


We can see the graph does not go beyond -1 on the x axis on the left and not beyond 2 on the right

The domain -1 $\leq $ x $\leq $ 2

It can also be written as [-1, 2]

The range we look at the smallest y value, which is 1 and the highest value is 10

The range 1 $\leq $ y $\leq $ 10

In the interval notation it can be written as [1,10]

Example 2:

Find the domain and range from the given graph

Function Example


Looking left to right, we see the first x value is at -5 and we see even though the graph is broken, the y value exist at the broken part of the curve so x is still defined till x=9 which is denoted by a hollow circle which means x is not included.

The domain is [-5, 9)

Range : we can see that the graph does not cross 3 and the lowest point is y=1.

The range [1, 3]
How can you tell whether a graph is the graph of a function?

First we need to understand the definition of the function which says that for every x there is exactly one y value. To check if the given graph is a function, one just need to apply the vertical line test. If at any x value the line passes through 2 or more point on the curve then the given graphs is not the graph of a function.
Example 3:

Is the given graph a function?

Function Examples

NO, when we draw a vertical line it intersects at more than one point.

Function Problem
Example 4:

Is the graph a graph of a function
Function Problems



When we draw the vertical line it intersects at only one point

Function Solution

More topics in  Function
Properties of Functions
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