If you've been comfortable working with equations like y = 3x + 10, you already understand function notation. You just don't know it yet.
It's the same as y =, with a new name
When you see f(x) = 3x + 10, it means the exact same thing as y = 3x + 10. The f(x) is just a different label for the output. Instead of calling it y, we're calling it “f of x” — which is a way of saying “the output of function f when the input is x.”
f(x) is just another name for y. Same equation, different label.
The reason math uses this notation is that sometimes you're working with more than one equation at once. If you have two different relationships, calling them both y gets confusing. So instead, you call one f(x) and the other g(x). They're still just equations with inputs and outputs — they just have different names so you can keep track of which is which.
What “f(3) = 19” means
When you see f(3) = 19, it's telling you: “when you plug 3 into the function, the output is 19.” In y-equation terms, that's the same as saying when x = 3, y = 19. It's just a more compact way of writing it.
So if f(x) = 5x + 4, and someone asks you to find f(3), you plug in 3 wherever you see x:
That's all “evaluating a function” means — substituting a number in and calculating.