What does function mean in mathematics?

[abhijath]

The distance covered by a car at time t=0 is 0 / at time t= 1 is 10 km/ at time t= 5 is 50 km,

With this example can you explain, what is function of what? f(d) or f(t),

 

[mfb]

The question does not make sense.

Distance as function of time can be formulated as function, time as function of distance can be written down as a function as well.
We can also find your post count as function of time, which should be 1 as you posted here, @Greg Bernhardt can check what happened.

 

[HallsofIvy]

At a given time, the automobile will be at a specific distance at any time so distance, d, will be a function of time, t. If it happens that the automobile continues in a straight line so it is never at the same distance for two different times, then time, t, is also a function of distance, d.

 

[abhijath]

At a given time, the automobile will be at a specific distance at any time so distance, d, will be a function of time, t. If it happens that the automobile continues in a straight line so it is never at the same distance for two different times, then time, t, is also a function of distance, d.

thanks for your reply sir,

ok so it means that d=f(t) and also t=f(d), now as you said if distance is never same for two different times then t=f(d), but wat if the car comes to halt at time t=6, wether still time can be said as function of distance t=f(d)?

 

[mfb]

It's hard to read your posts.

You should not use the same name ("f") for two different functions.

but wat if the car comes to halt at time t=6, wether still time can be said as function of distance t=f(d)?

No, as the distance is the same for two different points in time.

 

[HallsofIvy]

You titled this "What does function mean in mathematics so I was using a mathematical definition of "function"- a set of ordered pairs, {(x, y)}, such that no two different pairs have the same "x"- that is (3, 4) and (3, 5) could not appear in the same function. If we write y= f(x) we want to have some way of determining what y must be for any given x. We cannot have f(3)= 4 and f(3)= 5 (though we might well have f(3)= 5 and f(4)= 5).

 

[Ocata]

You might have the functions mixed up. If you want to choose a time and see what the distance will be, the function will look like this d = f(t). If you want to choose a distance and see what the time will be, the function will look like this: t = f(d) or f(d) = t. Same thing.

For instance, with your original example, your function is probably: d = f(t), where f(t) = 10t, so you have [d = f(t)] => [1 = 10(0)] => [10 = 10(1)] => [50 = 10(5)]

for choosing a distance and finding what the time will be:

[d = 10t] => [d/10 = t], so the function will be: [g(d) = t] => [(1/10)d = t] => [(1/10)50 = 5] => [(1/10)10 = 1] => and then you do the same thing to find the time when the distance is zero by plugging in 0 for d in the g function => g(0) = 0

 

[Edwin McCravy]

In complex analysis there are multi-valued functions. So there seems to be a conflict in the terminology.

 

[aikismos]

In complex analysis there are multi-valued functions. So there seems to be a conflict in the terminology.

According to WP, a misnomer (See https://en.wikipedia.org/wiki/Multivalued_function.)

'In the strict sense, a "well-defined" function associates one, and only one, output to any particular input. The term "multivalued function" is, therefore, a misnomer because functions are single-valued.'

But it also suggests the term has fallen out of use.

'The practice of allowing function in mathematics to mean also multivalued function dropped out of usage at some point in the first half of the twentieth century. Some evolution can be seen in different editions of A Course of Pure Mathematics by G. H. Hardy, for example. It probably persisted longest in the theory of special functions, for its occasional convenience.'

 

[mfb]

You can find a proper function for every multivalued function if you set its codomain to the powerset instead of the original set.
As an example, for the complex logarithm you would get ln(1)= {0, 2pi i, - 2 pi i, ... }.

 

[Edwin McCravy]

Thanks. I like that idea of considering the set of all possible outputs as a single output. I studied complex analysis years ago before mathematicians were concerned with such as that. This was back when topology was rubber-sheet geometry, and 1 was considered prime. (Yes I'm old!) :)

 

[HallsofIvy]

In complex analysis there are multi-valued functions. So there seems to be a conflict in the terminology.

Yes, but the question was clearly referring to real numbers

 

[Edwin McCravy]

From what I can find online, a function is a relation from a set of inputs (the domain) to a set of possible outputs (the codomain) where each input is related to exactly one output. I can find no mention of any requirement that the inputs and outputs of a function must necessarily be real numbers (or even numbers at all!) What about the "range"? Is the range necessarily the same as, or only a subset of, the codomain?) One professor I had in grad school back in the 60's always spoke of "the set of departures" and "the set of arrivals". I only taught undergraduate mathematics, col alg, trig, discrete, three calculi and diffy-Q, so all this extreme precision in terminology was no big deal. I often wonder who gets to decide mathematical terminology anyway, or is it really cut-and-dried?

.

 

[mfb]

I can find no mention of any requirement that the inputs and outputs of a function must necessarily be real numbers (or even numbers at all!)

They don't have to be, but times and distances (see post 1) are usuallly expressed with real numbers.

What about the "range"? Is the range necessarily the same as, or only a subset of, the codomain?)

It can be a (nontrivial) subset.

I often wonder who gets to decide mathematical terminology anyway, or is it really cut-and-dried?

Community consensus.