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Function

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.

 Related Calculators Calculator Functions Calculate Exponential Function Calculate Inverse Function calculating gamma function

Examples

Example 1:

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

Solution:

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

Solution:

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?

Solution:

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

Example 4:

Is the graph a graph of a function

Solution:

Yes

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

 More topics in  Function Properties of Functions
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