# What does “x as a function of y” really mean?

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I learned functions just purely as a mathematical concept, but I realised as I studied physics along with it, that it is actually a fundamental concept in the world. But I never really understood the phrase. For example, ‘velocity as a function of time’, yes I get that this would mean a v=tx form of graph, but what does it actually *mean*? Thank you

Edit: I made a mistake previously. I accidentally said ‘time as a function of velocity’

In: Mathematics

If x is a function of y, it means that the value of x is controlled by the value of y and only one (or no) value of x is possible for any given value of y.

it means it depends on it. So my stride length while walking is a function of the length of my legs. When I cook the dryness of what I bake is often a function of the amount of flour in something.

If I had to write a formula then that would be my changing variable, everything else is constant (in the real world things are a function of many things). The speed my car goes at is a function of my pressing on the accelerator, as well as the incline at which I’m travelling.

So for the stereotypical physics/math problem, “two trains heading towards eachother, when do they meet?” is a function of the speed of each train, and the angle of intercept between their courses.

Does that make sense? we could work through an example if you want to look around your room and pick something to explain (the brightness of your screen is function of how much power it is using)

‘time as a function of velocity’ is backward. It should be ‘velocity as a function of time’. What that means is, you have an equations where time is the variable and will tell you what the velocity is at any given time. With that knowledge, you could also plot position as a function of time. With these two things, you could tell the position and speed of the object over time.

Functions simply describe relations between different quantities. Specifically, it’s something that you have a set of inputs for, it performs operations on them and then produces some output value. The phrase “Y as a function of X” simply states which quantities are outputs (*Y* here) and which are inputs (*X*).

“velocity as a function of time” would be something like *v*(*t*). Regardless of what the specific function looks like, it indicates that you’re interested in the velocity *v* and you know the time *t*. This can be relevant if you’re dropping something, or accelerating in a race.

It’s also possible to to have the inverse: “time as function of velocity”, *t*(*v*). In that case you’re saying that you know the velocity and are interested in the time. This can be relevant if you’re throwing something and want to know how long it takes for it to hit the ground again.