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

In summary, there is a type of function known as 'multi-valued' where a single input value can be mapped onto multiple output values. However, this is not considered a valid function in mathematics. The term 'function' is the correct term for this concept, not 'equation'. There are also one-to-one functions where a single input value is mapped onto a single output value, and these must pass both the horizontal and vertical line tests. It is important to use the correct terminology when discussing mathematical concepts.
  • #1
Chrisistaken
12
0
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
 
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  • #2
These functions are known as 'multi-valued'.
 
  • #3
Chrisistaken said:
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 [itex]\sqrt{x}[/itex] to be the positive number whose square is equal to x. What is NOT a "function of x" is the y given by [itex]x= y^2[/itex]. For example, while solutions to the equation [itex]y^2= 4[/itex] are 2 and -2, [itex]y= \sqrt{4}[/itex] 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, [itex]\sqrt{-4}[/itex] would be both -2i and 2i.
 
  • #4
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
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
 

1. What is a function with more than one y value for a given x value?

A function with more than one y value for a given x value is known as a multi-valued or multi-valued function. This means that for every input value (x), there is more than one output value (y) produced by the function.

2. Why do some functions have more than one y value for a given x value?

This can occur when the function is not one-to-one, meaning that two different input values can produce the same output value. This often happens when the function is not a straight line and has curves or bends.

3. How can we graph a function with more than one y value for a given x value?

When graphing a multi-valued function, we can plot each output value as a separate point on the graph. This will result in multiple points for the same input value on the graph. Alternatively, we can use different colors or symbols to represent each output value on the graph.

4. Can a function with more than one y value for a given x value still be considered a function?

Yes, a function with more than one y value for a given x value is still considered a function as long as each input value (x) corresponds to exactly one output value (y). This means that the function is still well-defined and follows the vertical line test.

5. How can we determine the domain and range of a function with more than one y value for a given x value?

The domain of a function with more than one y value for a given x value is all the possible input values (x) for the function. The range is all the possible output values (y) for the function. To determine the domain and range, we can look at the graph of the function or use algebraic methods to find the values that satisfy the function.

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