Functions

In math, a relation is a pairing of inputs and outputs.  Functions normally use equations to do something to the input to get the output, normally called the range or variable y, and functions don't need to be linear when graphed. Functions are a specific category of relations, and all functions are relations. A function is a relation that has exactly one output paired with one input, but not necessarily one input for each output. For example, a function would be walking. A good way to think of it is if you walk to the north, you only go north, you walk south, you only go south, that is a function because you can predict where you go. Your input, the direction you walk, is correlated with the output, the direction you go. However,  All the possible real values of x that can be used to get a real and defined output y is called the domain of the function, and all the real, defined, and possible values of y with a real input x is called the range of the function.

How do we solve functions, whatever they want us to do?

Normally, when given a question involving a function, you use the function to get an output or find what the input was for an output. Another thing questions may ask you to do is find the equation that turns an input into an output, which they give you. Here are some examples of function problems:

1. A function using the equation y = -3(7x+2). For input x = -2, what is y?

2. A function using the equation y = 6x. If an output was 18, what was input x?

3. A function had input x, and output y. If x = 2 and y = 8, what was the equation?

Now we have the questions, but how do we solve them? Here are the solutions corresponding to the problems.

1. We first put x into the equation and then simplify. -3(7*(-2) + 2) ---> -3(-12) ---> 36. So, for input -2, we have output 36. 

2. We put 18 as y into the equation, so 18=6x, and we solve, getting x = 3

3. x=2, y=8. Now, the first step is to try and divide y by x. Normally, you would be given more than one pair of inputs and outputs, but this is simple, and y/x is 4, so y = 4x.

Function problems: (Solve everything for 3 KDC)