Math CentralQuandaries & Queries


Question from Farzan, a student:

Is x/y + y/x = 2 a function or not.

Hi Farzan,

First note that neither x nor y can be zero.

Multiply both sides of the equation by xy. Simplify and factor. You will obtain a much simpler algebraic expression, one that has a graph which is familiar (but remember x ≠ 0).

I hope this helps,

Farzan replied:

Dear friends, In my last question penny answered that we should multiply both sides by xy, but it is wrong, our teacher said we can't do this.


Why not? It is mathematically logical and correct. You need to be careful that you don't multiply or divide by zero, but that isn't the case here.

Here's proof:

Statement Reason
x/y + y/x = 2 Given.
      x≠0 and y≠0 Because then the original question would be dividing by zero.
      xy≠0 Because neither factor is zero.
(xy) (x/y + y/x) = (xy) 2 Multiply both sides of given equation by (xy); we can do this because we are not multiplying by zero.
x2 + y2 = 2xy Simplify.
x2 - 2xy + y2 = 0 Make one side zero.
(x-y)2 = 0 Factor it.
(x-y) = 0 Take the square root of both sides.
x = y. Solved.

But remember that the caveats still apply: neither x nor y is allowed to be zero.

Given this, do you have a function or not? A function is a relationship in which each permitted value of x has a unique value of y corresponding to it. In other words, if x1 ≠ x2, then ƒ(x1) ≠ ƒ(x2).

Stephen La Rocque.

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