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A function relation
Functions are mathematical representations of input-output relationships. These can be as simple as plugging $$x=2$$ into $$3x+4$$ to get $$10$$, but we also encounter many of these functional relationships in our everyday lives. For example, the distance a car can drive is a function of how many gallons (or liters) of gasoline are put into it. The function of a car that can drive 15 miles on 1 gallon of gasoline would be $$f\left(x\right)=15x$$. In this function, $$x$$ is the domain, or input, of the function and represents the number of gallons of gasoline put into the car. $$f\left(x\right)$$ is the range, or output, of the function and represents the distance in miles (or kilometers) that the car can travel.

There are some limits to this function, however. It is impossible to fill the gas tank with less than zero gallons of gasoline and we can't fill the gas tank with more than it can hold. We also can't fill it with anything other than gasoline or it will not drive. In the function, this means that $$x$$ must be larger than zero, smaller than the volume of the car's gas tank, and represent only gasoline. The domain of the function does not cover all possibilities; there are limits to what can be plugged into this function. The same goes for range, the function's output. It is impossible for the car to travel less than zero miles (or kilometers) and it can't travel more than 15 times the capacity of its gas tank.

Every function has a set of possible inputs called the domain and a set of possible outputs called the range. These can be infinite, exclude specific numbers, be only positive, or include other types of conditions. What is true of all functions, however, is that their inputs each have exactly one output. Any more or less would mean it was not a function.

In order to understand a function, we need to know its domain and range.