For our introduction in Functions I, I recommend thinking of a function as a machine that takes an INPUT and then spits out an OUTPUT. Functions are equations denoted by the $f(x)$ symbol. Let's make sure we understand all of the terminology with an example.
$$f(x) = x^2 +1$$
The input is what's between the parentheses. In this case, the input is simply $x$.
The output is $f(x)$, which is also equal to $x^2+1$, so both of these can be considered the output. Let's recap:
- INPUT: $x$
- OUTPUT: Either $f(x)$ or $x^2+1$.
So let's imagine we start inputting values into our function machine, say $x=3$. We then get...
$$f(3)=3^2 +1 = 10$$
So we have $f(3) = 10$. In other words, when the input is $3$, the output, $f(3)$, is $10$.
Another Thing to Keep in Mind
You can set one function equal to another. For example...
$$f(x) = f(x+1)$$
What exactly is this saying? It's saying that the output of the left side of the equation, $f(x)$ is equal to the output of right side of the equation, $f(x+1)$, even when we add $1$ to the input. Remember the input is what's inside the parentheses. So let's translate...
Even when we add $1$ to the input, the output remains the same. So if $f(3) = 7$, we also know that $f(4)=7$. And $f(5)$..and $f(6)$...you get the idea.