Which of the following equations have the same graphs? $(A)y=x-2\,(B)y=\frac{x^2-4}{x+2}\,(C)(x+2)y=x^2-4$

Question

Which of the following equations have the same graphs?
$1.y=x-2\hspace{1cm}2.y=\frac{x^2-4}{x+2}\hspace{1cm}3.(x+2)y=x^2-4$

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I know that the graph of first and the second functions are different because in the second function,there will be a hole at $x=-2$ whereas the graph of the first function will be a smooth graph with no holes.

But the book answer says that all the equations have different graphs.

I do not understand how the graphs of the second and the third equations are different because i thought the graphs of the second and the third equations are same.

Please help me.Thanks.

Answer 1

Rewrite (C) as $(x+2)(y-x+2)=0$. Then you see $(x,y)$ is a solution to (C) if $y=x-2$ or if $x=-2$.

So (C) is different from both (A) and (B).

Answer 2

1) and 2) are functions. No x has more then 1 y associated with it

3) is not a function. At x = -2, there are an infinite number of y's associated with x = -2. (Because 0*y = 0 is true for all y.)

1) and 2) are functions and they are the same everywhere except at x = -2 where 2) is undefined and "has a hole".

So they are all different.