A function describes a relationship between two or more variables. Consider the equation `y=x+1`. This equation describes a relationship between the variable `y` and the variable `x`, and it tells you that the value of `y` is always `1` more than the value of `x`.
The syntax `f(x)` tells you that `f` is a function, and it takes `x` as an input. `f(x)` defines a relationship between two things that are represented mathematically by variables, `x` and `y`, where `y=f(x)`. I have seen many students confused by the function notation, `f(x)`. I think it's just a result of not having much experience with functions, or maybe having a previous teacher who wasn't good at explaining what they are, or both.
Anyway, when you see `f(x)` it is pronounced, "`f` of `x`". The `f` and the `x` are not separate; it's not `f` times `x` as some students want to think. They are one thing that defines a relationship between a variable `x` and a variable `y`. Where is `y`, you say? `y` comes from evaluating the function, `f(x)`, at specific values of `x`; `y` is the output of the function. So `y` is `f(x)`. Sometimes you will see just the `f` without the `(x)` part. Don't think that's something different; it's not. The `f` by itself refers to the function in general; the set of operations that are performed on the input to the function, whatever that input happens to be. `f(x)` refers to an instance of the function, `f`. A function has to have an input, and the input is whatever is inside the parentheses. Let's go through a couple of examples and hopefully, this will make more sense.
Suppose you have a function, `f(x)=x^2 + 2x + 1`. This takes `x` as its input (what's inside the parentheses) and the right-hand side of the equation defines what the function does to the input, in this case, `x`. When you see this function, you should think:
Now consider the function, `g(x)=sqrt(4x+7)`. Again the input to this function, `g`, is just `x`. With this function you should think:
Another confusing function thing for many people is when we have an instance of a function like, `f(x+1)`. In general, this is not equal to `f(x) + 1`. Consider the function in the first example above, `f(x) = x^2 + 2x + 1`. Remember that whatever's inside the parentheses in the function notation defines what the input to the function is, and the right-hand side of the equation defines what the function does to the input. It might be clearer to write the function this way,
Consider the linear function, `f(x) = 5x + 3`. Here's what we know about this function just from the equation for `f(x)`.
Consider the quadratic function, `f(x)=3x^2 - 6x - 9`. Here's what we know about this function from the equation for `f(x)`.
Consider the cubic function, `f(x)=x^3 - 3x^2 - 33x + 35`. Here's what we know about this function from the equation for `f(x)`.
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