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Welcome to our Math lesson on **Function or not a Function? The Vertical Line Test**, this is the seventh lesson of our suite of math lessons covering the topic of **Injective, Surjective and Bijective Functions. Graphs of Functions**, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

In tutorial 13.3 "Modelling Curves using Logarithms", there is a paragraph named "A Brief Introduction to Functions", where we have anticipated the definition of a function and its graph. The following passage about this concept is excerpted from that tutorial:

"A good method for checking whether a given graph represents a function or not is to draw a vertical line in the sections where you have doubts that an x-value may have in correspondence two or more y-values. If the vertical line intercepts the graph at more than one point, that graph does not represent a function."

This feature which allows us to check whether a graph belongs to a function or not, is called the "**vertical line test**". Not all relations are functions, as we saw in the previous tutorial. Therefore, since every relation between two quantities x and y can be mapped on the XOY coordinates system, the same x-value may have in correspondence two different y-values - points which when shown in a graph, lie in the same horizontal position (the same x-coordinate) but at two different heights (different y-coordinates). Look at the graph of the relation below.

It is obvious that if we draw a vertical line from anywhere on the right of -1, it intercepts the graph at two different points. One of these lines could be the vertical axis; however, we will draw another vertical line to better highlight this feature. For example, if we draw a vertical line at x = 2 we obtain two y-intercepts with the relation's graph: y_{1} = -3.873 and y^{2} = 3.873, as shown in the figure below.

The above y-intercepts are found by substituting x = 2 in the original relation and calculating the corresponding y-values. Thus,

y^{2} = 2^{3} + 2 ∙ 2 + 3

= 8 + 4 + 3

= 15

= 8 + 4 + 3

= 15

Therefore, since y = √15, we obtain

y_{1} = -√15

= -3.87298

≈ -3.873

= -3.87298

≈ -3.873

and

y_{2} = √15

= 3.87298

≈ 3.873

= 3.87298

≈ 3.873

Use the vertical line test to determine whether the following graphs represent functions or not.

- This is not a function as, if for example, we draw a vertical line at x = -3, this line intercepts the graph at two different points A and B, as shown in the figure. The y-intercepts of A and B are: Hence, the two intercepts are: A(-3, -0.507) and B(-3, 0.507).
*1**/**3*^{2}+= 4*1**/**y*^{2}+*1**/**9*= 4*1**/**y*^{2}= 4 -*1**/**y*^{2}*1**/**9*=*1**/**y*^{2}-*36**/**9**1**/**9*=*1**/**y*^{2}*35**/**9*

y^{2}=*9**/**35*

y = √*9**/**35*

= ±√*9**/**√35*

=*±3**/**5.916*

= ±0.507 - This is the graph of a circle with centre at C(1, 0) and radius 3 units. Therefore, if we draw a vertical line through the centre (x = 1), it intercepts the circle at two different points A and B, as shown in the figure. The two y-intercepts are found by substituting x = 1 in the formula of the original relation. Thus, (1-1)Therefore, the two intercepts are: A(1, -3) and B(1, 3).
^{2}+ y^{2}= 9

0^{2}+ y^{2}= 9

y^{2}= 9

y = √9

y = ±3 - This is the graph of a polynomial function, as the original expression it can be written in the form 4y = xor
^{3}+ 3x^{2}- 1y =The vertical line test cannot identify any part of the graph in which there are two y-coordinates for a single x-coordinate, despite the multiple attempts in this regard, as shown in the figure below. Therefore, the vertical test cannot identify any point of the graph with two y-coordinates for the same x-coordinate. Hence, this is a function.x*1**/**4*^{3}+x*3**/**4*^{2}-*1**/**4*

You have reached the end of Math lesson **16.2.7 Function or not a Function? The Vertical Line Test**. There are 7 lessons in this physics tutorial covering **Injective, Surjective and Bijective Functions. Graphs of Functions**, you can access all the lessons from this tutorial below.

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