Function notation looks intimidating — $f(x)$, $f(g(x))$, $f^{-1}(x)$ — but it's just a labeling system that lets you talk about functions cleanly. Once you know what each symbol *means*, the questions become procedural.
A function is a rule that takes an input and gives back exactly one output. We label functions with letters (usually f, g, h) and write f(x)= [the rule] to define one.
f(x)=2x+3 means: whatever you plug in for x, multiply by 2 and add 3.
Reading function notation:
f(5) = the output when you plug 5 in for x.
f(a+1) = the output when you plug (a+1) in for x. Substitute exactly what's inside the parentheses.
f(g(x)) = a composition. Compute g(x) first, then plug that into f.
Plugging in is the key skill. When you see f(x+2), replace EVERY x in the function's rule with (x+2) — including the parentheses. If f(x)=x2+3x, then
f(x+2)=(x+2)2+3(x+2)
Composition: f(g(x)). Read it inside-out. If f(x)=x+1 and g(x)=2x:
Compute g(x)=2x first.
Plug that into f: f(2x)=2x+1.
So f(g(x))=2x+1.
Note: f(g(x)) and g(f(x)) are usually different! Order matters.
Reading from a graph or table:
"Find f(3) from this graph" → look at x=3, read off the y-value.
"Find x when f(x)=5" → look at y=5, read off the x-value (could be more than one).
Inverse functions f−1(x): the function that undoesf. If f(2)=7, then f−1(7)=2. To find f−1 algebraically: write y=f(x), swap x and y, then solve for y.
If f(x)=x2−2x, what is f(a+3)?
If f(x)=3x−2 and g(x)=x2, what is f(g(2))?
f(x+2) with f(x)=x2 is (x+2)2, NOT x2+2 or x+22. Always wrap the substitution in parentheses, expand carefully, then simplify.
f(x)⋅g(x) is multiplication. f(g(x)) is composition (plug g's output into f). They're completely different operations.
f(g(x)) is read inside-out: g(x) first, then f. Don't compute f(x) and then plug into g — that's g(f(x)), which usually gives a different answer.
f(2) means evaluate f at input 2 — substitute. f⋅2 means multiply f by 2 — different. The parentheses indicate function evaluation, not multiplication.
f(x) is a function — a rule mapping each input to one output. The letter (f, g, h) is just the function's label.
Substitution: replace every x in the rule with whatever's inside the parentheses, using parentheses to keep order of operations.
Composition f(g(x)) is inside-out: g first, then f. Order matters.
From a graph: f(a) is the y-value at x=a. To find x when f(x)=b, look at y=b and read off the x.