Function notation

Why f(x) is a machine's output, not a multiplication — evaluating f(3), reading f(x) = k as an equation to solve, and keeping inputs and outputs straight.

Algebra · Functions

Function notation

Why f(x) is a machine's output, not a multiplication — evaluating f(3), reading f(x) = k as an equation to solve, and keeping inputs and outputs straight.

Why it works

f(x) = 3x - 4

names a machine: feed in a number

x

, and out comes

3x - 4

. The notation

f(3)

means "the machine's output when the input is 3" — it is NOT

f \times 3

; the brackets here hold an input, not a multiplication.

Evaluating: substitute the input everywhere $x$ appears.

f(5) = 3(5) - 4 = 11, \qquad f(-2) = 3(-2) - 4 = -10.

All the substitution discipline carries over — brackets round negatives, powers before coefficients: with

g(x) = x^2 + 1

g(-3) = (-3)^2 + 1 = 10

.

$f(x) = 20$ is an EQUATION — solve it. It asks: *which input gives output 20?*

3x - 4 = 20 \;\Rightarrow\; x = 8.

The two directions are opposites and exams test both ways:

f(20)

feeds 20 IN;

f(x) = 20

asks what produces 20 OUT. Read which way the question points before touching the algebra.

Quadratic machines can have two inputs for one output.

g(x) = 50

means

x^2 + 1 = 50

, so

x^2 = 49

and

x = \pm 7

— BOTH inputs work, and both are the answer. A square in the formula is the cue to expect two.

$f(x) = x$ is a meeting of input and output. Solving

5 - 2x = x

finds the input the machine leaves unchanged (

x = \frac{5}{3}

) — just an equation like any other, once written out.

Different letters, same idea:

g

h

\text{f}

are separate machines living in the same question; each keeps its own formula.