You know that a function gives exactly one output for each input. Function notation is a compact way to say which input you mean. If $f(x) = 2x + 3$, then $f(4)$ is the output when the input is $4$: $f(4) = 2(4) + 3 = 11$. It is evaluating an expression, with a name attached.
Why it works
The letter $f$ names a rule, and the parentheses hold the input. So $f(4)$ is read “$f$ of $4$,” not “$f$ times $4$.” Naming functions lets us talk about several at once ($f$, $g$, $h$) and write sentences like $f(4) = 11$, which says “input $4$ gives output $11$” in one short equation. That is also a point on the graph: $(4, 11)$.
Key idea
$f(a)$ means: replace every $x$ in the rule for $f$ with $a$, then simplify. The result is the output, and $(a, f(a))$ is a point on the graph of $f$.
Worked examples
Worked example: Evaluate at a negative input
If $g(x) = x^2 - 5x + 1$, find $g(-2)$.
- Replace every $x$ with $(-2)$: $g(-2) = (-2)^2 - 5(-2) + 1$.
- Simplify: $4 + 10 + 1 = 15$.
Answer: $g(-2) = 15$
Worked example: Combine outputs
If $f(x) = 3x - 4$, find $f(2) + f(-1)$.
- $f(2) = 3(2) - 4 = 2$.
- $f(-1) = 3(-1) - 4 = -7$.
- Add the outputs: $2 + (-7) = -5$.
Answer: $-5$
Worked example: An expression as the input
If $f(x) = x^2 + 1$, find $f(a + 2)$.
- The input is the whole expression $a + 2$, so put it in parentheses: $f(a + 2) = (a + 2)^2 + 1$.
- Expand: $a^2 + 4a + 4 + 1 = a^2 + 4a + 5$.
Answer: $a^2 + 4a + 5$
Common mistake
Reading $f(x)$ as “$f$ times $x$” and computing $f(3)$ as $3f$. There is no number called $f$ to multiply; $f$ is the name of a rule. Check: $f(3)$ must be a single number you get by putting $3$ into the rule.