Math Help [Request Topics]

Functions and Graphing

Functions work like a machine. You put something in, and you get something out. In our case, what we put in is a variable x. This variable is typically known as the independent variable because it does not depend on other parameters.  Up until this point, you are probably used to seeing equations written in the following form:

y = (some expression in terms of x) 

For example, y = x^2, y = mx + b,  y = ax^2 + bx + c, etc.

But now, we are going to write equations in a new form:

f(x) = (some expression in terms of x)

In this form, f(x) takes the place of y. We normally read this as "F of x equals ...." We call f(x) our dependent variable because the value we get for f(x) depends on the value for x that we put in. I'll give a diagram of a simple example to show what I mean for the function f(x) = x^2.

There are some rules when it comes to functions. One important rule is that a single input to a function cannot have multiple outputs. What this means is that if I input a value into f(x), I can only get one value out. However, a single output can have multiple inputs. This means that I can have multiple values of x have the same f(x) value. A good way to think about this is like mapping values from one map to another map. I've drawn a diagram below to show you what I mean. The two large circles are labeled as our input values x and the output values f(x). The small circles inside the large circles represent specific values. An arrow maps the value from x to a value in f(x).

As you can see, you are not allowed to have on x value lead to multiple f(x) values, but you can have multiple x values lead to one f(x) value. This sometimes leads to some complicated situations where if your teacher asks you to solve for x when f(x) = 0 or some numerical values, you can sometimes have multiple answers. Does this mean that only one of them is correct? No! In fact, the correct answer is to give all the values of x that solve the problem. Let's keep this simple for now by solving an easy example.