Example: Find out which of the following equations functions are & which are not functions.
y= 5x + 1
The "working" definition of function is saying is that if we take all of possible values of x & plug them in the equation & solve for y we will get accurately one value for each value of x. At this stage it can be pretty hard to actually illustrate that an equation is a function thus we'll mostly talk our way through it. Conversely it's frequently quite easy to show that an equation isn't a function.
So, we need to illustrate that no matter what x we plug in the equation & solve for y we will only obtain a single value of y. Note as well that the value of y will probably be different for each value of x, although it doesn't have to be.
Let's begin by plugging in some of the values of x and see what happens.
x= -4 : y= 5 ( -4) + 1 = -20 + 1 = -19
x= 0: y= 5 (0)+ 1 = 0 + 1 = 1
x= 10 : y= 5 (10) + 1 = 50 + 1= 51
Thus, for each value of x we obtained a single value of y out of the equation. Now, it isn't enough to claim that this is a function. To officially prove that it is a function we have to illustrates that this will work no matter that value of x we plug into the equation.
Certainly we can't plug all possible value of x in the equation. That just isn't possible physically. For each x, on plugging in, first we multiplied the x by 5 and after that added 1 onto it. Now, if we multiply any number by 5 we will obtain a single value from the multiplication. Similarly, we will only get a single value if we add 1 onto a number. So, it seems plausible that depend on the operations involved with plugging x into the equation that we will just get a single value of y out of the equation.
Hence, this equation is a function.