HOW TO DETERMINE A FUNCTION FROM A GRAPH

An effective tool that determines a function from a graph is "Vertical line test" 

The following are the steps of vertical line test : 

Step 1 : 

Draw a vertical line at any where on the given graph. 

Step 2 : 

We have to check whether the vertical line drawn on the graph intersects the graph in at most one point.

Step 3 : 

If the vertical line intersects the graph in at most one point, then the given graph represents a function. 

If the vertical line intersects the graph in more than one point, then the given graph does not represent a function. 

Caution : 

For some graphs, the vertical line will intersect the graph in one point at one position and more than one point at a different position.

In the above situation, the graph will not represent a function. 

Key Concept :

A graph represents a function only if every vertical line intersects the graph in at most one point.

We can have better understanding on vertical line test for functions through the following examples. 

Example 1 : 

Use the vertical line test to determine whether the following graph represents a function.

Solution : 

Step 1 : 

Draw a vertical line at any where on the given graph. 

Step 2 : 

We have to check whether the vertical line drawn on the graph intersects the graph in at most one point.

Step 3 : 

In the above graph, the vertical line intersects the graph in at most one point, then the given graph represents a function. 

Example 2 : 

Use the vertical line test to determine whether the following graph represents a function.

Solution : 

Step 1 : 

Draw a vertical line at any where on the given graph. 

Step 2 : 

We have to check whether the vertical line drawn on the graph intersects the graph in at most one point.

Step 3 : 

In the above graph, the vertical line intersects the graph in more than one point (three points), then the given graph does not represent a function. 

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