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

1. Oct 16, 2011

### 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

Last edited: Oct 16, 2011
2. Oct 16, 2011

### SteamKing

Staff Emeritus
These functions are known as 'multi-valued'.

3. Oct 16, 2011

### HallsofIvy

'something-or-other'= function.

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.

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.

4. Oct 16, 2011

### 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.)

5. Oct 17, 2011

### 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.