Functions having more than one y value for a given x value

[Chrisistaken]

Hi,

In school (I think) I recall there being a test for an equation which determined whether or not it was a valid 'something-or-other' and it was simply that if you could draw a vertical line anywhere on the graph of the equation, that crossed the line more than once, it was not a valid 'something-or-other'.

An example of an invalid equation would be: y = sqrt(x)

Does anyone know what the 'something-or-other' is? And if there is a term for these equations, what is it?

My apologies if this was just a figment of my imagination.

Regards,

Chris

 

[SteamKing]

These functions are known as 'multi-valued'.

 

[HallsofIvy]

Hi,

In school (I think) I recall there being a test for an equation which determined whether or not it was a valid 'something-or-other' and it was simply that if you could draw a vertical line anywhere on the graph of the equation, that crossed the line more than once, it was not a valid 'something-or-other'.

'something-or-other'= function.

An example of an invalid equation would be: y = sqrt(x)

No, it isn't. In order that it be a function, we specifically define $\sqrt{x}$ to be the positive number whose square is equal to x. What is NOT a "function of x" is the y given by $x= y^2$. For example, while solutions to the equation $y^2= 4$ are 2 and -2, $y= \sqrt{4}$ is 2 only.

Does anyone know what the 'something-or-other' is? And if there is a term for these equations, what is it?

My apologies if this was just a figment of my imagination.

Regards,

Chris

In complex numbers, where there turn out to be practically NO such "functions", we allow what we now call "multivalued functions". In that case, $\sqrt{-4}$ would be both -2i and 2i.

 

[eumyang]

If I may suggest, please fix your title. If we are defining functions such that the x-value is the input value and the y-value is the output value, then the title does not make sense. If you have a single x-value mapped onto multiple y-values, then you wouldn't have a function in the first place.

Instead, you should have used the word "relation" in your title. In relations, a single input value can be mapped onto multiple output values. All functions are relations, but not all relations are functions.

There are also a specific type of function called "one-to-one function." In functions, it's permitted for multiple input values to be mapped onto a single output value. (y = x² is an example. Except for x = 0, there are always two x-coordinates with the same y-coordinate, like (-7, 49) and (7, 49).) In one-to-one functions, however, you have a single input value mapped onto a single output value. Graphically, they must pass both the horizontal and vertical line tests. (Easiest example is y = x.)

 

[Chrisistaken]

Thanks all for your timely replies, much appreciated and my apologies to eumyang for the misleading title. I had a sneaking suspicion that the "something-or-other" I was searching for was infact "function" and had gone through my question at the last moment prior to posting, to change "function" for "equation". Must've overlooked the title.

Well anyway, once again thanks for the answers, most helpful :)

Regards,

Chris