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Question 541731: can someone please help me?
tell whether the relation is a function or not a function.
x=-4
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! This is NOT a function. The graph is a vertical line that goes through the point -4 on the x axis.
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A function can only have one unique value of y for a single given value of x. Your particular problem can have many values of y for the single given value of -4 for x. (Each point on that vertical graph has a corresponding different value of y for the single value of -4 for x.) Here are some points on the graph:
(-4,0)
(-4,-3)
(-4,+20)
(-4,-100)
(-4,+100)
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Notice that the value of x is always -4. Then notice that y can have many values for that single value of -4 for x. To be a function, y can only have one value for each value of x.
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Some more examples:
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Is y = 2x + 5 a function? Answer is YES. You can select various values for x and substitute them into 2x + 5. You never get two or more values of y for that particular value of x.
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Is x = y^2 a function? Answer is NO. Example. Let x = 4. That means that y^2 = 4 and if you take the square root of both sides to solve for y you get y = +2 and also y = -2. (In both cases when you square y you find that x = 4.) So y can have two values that correspond to x = 4. Since a function means that y can have only one value for each value of x, this means that x = y^2 cannot be a function.
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How about the graph of the ordered points:
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(4, 7)
(5, -3)
(-2, 10)
(0, 27)
(5, -8)
(27, 0)
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Does this graph that consists of a bunch of points represent a function? The answer is NO. Why? Because if you look at these points carefully you will see two points (5, -3) and (5, -8). That means that for the single value of x = 5, y can be either -3 or -8. Therefore, y has more than one value for x in that case and this is not permitted in a function.
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If we eliminated the point (5, -8) from the list so that the list was now:
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(4, 7)
(5, -3)
(-2, 10)
(0, 27)
(27, 0)
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Would the plot of these points represent a function. This time the answer is YES because for each value of x you only have a single corresponding value of Y.
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Another way to look at it is, on a graph of a function you can draw a vertical line anywhere along the x-axis and if that vertical line crosses two or more ordered points, then the graph cannot represent a function. In the above list of points that contained (5, -3) and (5, -8) a vertical line through x = 5 on the horizontal axis would cross through these two points. This again shows that the graph of all these points cannot be a function because a vertical line on this single plot crosses two points.
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Note this vertical line method does not apply for graphs of two functions. Imagine that you have the graphs of the two functions y = 3x + 7 and y = 5x - 2.
Obviously if you draw a vertical line it will cross the two functions. Does that mean that neither is a function? No. The vertical line is crossing two separate functions and x can have a single corresponding value of y on each of the two functions. That's OK. Both are separate functions.
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Hopefully this gives you a little more insight into what can and what cannot be a function.
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