⚙️ Function Kingdom · Functions

Function Notation

Read f(x) as a machine that turns an input into exactly one output, and evaluate and interpret function notation.

In short

f(a) means the output for input a — the brackets hold an input, never a multiplication.
A function can be shown as an equation, a table, a graph or a sentence, and all four say the same thing.
Evaluating substitutes a number for x; solving f(x) $= k$ works backwards and needs an equation.
Domain is the allowed inputs (watch square roots and division by zero); range is the possible outputs; composition works inside out.

A function is a machine

A function is a rule that turns each input into exactly one output. Put a number in, get a number out — and the same input always gives the same output.

The notation f(x) names the machine (f) and the input (x). So f(3) means "the output when the input is 3".

f(3) does not mean f times 3. The brackets hold the input, not a multiplication. Reading them as multiplication is the single biggest source of confusion in this topic.

To evaluate, replace every x in the rule by the input, using brackets if the input is negative:

f(x) $= 2x + 7 f(3) = 2(3) + 7 = 13 f(-4) = 2(-4) + 7 = -1$

The letter is arbitrary. f(t) $= 2t + 7$ is the same machine; only the name of the slot has changed.

Four ways to show the same function

The same function can be written in several ways, and moving between them is the real skill (design doc: multiple representations).

Equation: f(x) $= 2x + 3$

Table:

$x 0 1 2 3$ f(x) $3 5 7 9$

Graph: a straight line crossing the y-axis at 3 and rising 2 for every 1 across.

Words: "double the input, then add three."

Each view answers a different question quickly. The table makes the constant step of 2 obvious; the graph makes the steepness visible at a glance; the equation is the one you can compute with exactly.

In a table, the left column is the input and the right column is the output. Reading them the wrong way round is a frequent slip.

Going backwards: solving f(x) = k

Sometimes the output is known and the input is missing. That is no longer a substitution — it is an equation to solve.

f(x) $= 4x$ - 5, and f(x) $= 19 4x$ - $5 = 19 4x = 24 x = 6$

Notice the difference between the two questions. "Find f(19)" substitutes 19 for x and gives 71. "Solve f(x) $=$ 19" gives $x = 6$ . Same numbers, opposite directions.

In a table, the same reversal means scanning the output column for the value and reading back to the input column.

Domain, range and composition

The domain is the set of inputs the rule accepts; the range is the set of outputs it can produce.

Most rules accept everything, but two situations force a restriction:

square roots cannot take negatives, so f(x) $=$ sqrt(x - 3) needs $x >= 3$
division by zero is not allowed, so f(x) $= 1/(x$ - 2) excludes $x = 2$

Ranges are limited by the shape of the rule: f(x) $= x 2 + 5$ can never dip below 5, because $x 2$ is never negative.

Composition feeds one machine into another. f(g(x)) means: run g first, then push its output through f.

f(x) $= 2x + 1$ , g(x) $= x 2 f(g(3)) = f(9) = 19$

Order matters. Here $g(f(3)) = g(7) = 49, a$ completely different answer. Always work from the inside out.

Worked examples

Example 1

f(x) $= 3x 2$ - $5x + 2$ . Find f(-2).

Replace every x by -2, keeping brackets around the negative number.
$f(-2) = 3(-2) 2$ - $5(-2) + 2$ .
Deal with the power first: $(-2) 2 = 4$ , so the first term is $3 x 4 = 12$ .
$f(-2) = 12 + 10 + 2 = 24$ .

Example 2

f(x) $= 5x + 8$ . For which value of x is f(x) $= -7$ ?

The output is known, so set the rule equal to it: $5x + 8 = -7$ .
Undo the addition first: $5x = -15$ .
Divide both sides by $5: x = -3$ .
Check: $f(-3) = 5(-3) + 8 = -7$ . Correct.

Practice problems, with solutions

Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.

Problem 1

Difficulty 1 of 5

f(x) $= 3x + 6$ Find f(6).

Answer: 24

$f(6) = 3(6) + 6$
$= 18 + 6$
$f(6) = 24$

Problem 2

Difficulty 3 of 5

f(x) $= 2x + 14$ For which value of x is f(x) $= 14$ ?

Answer: 0

$2x + 14 = 14$
$2x = 0$
$x = 0 / 2$
$x = 0$

Problem 3

Difficulty 4 of 5

The table shows some values of a function f. The pattern continues. What is f(6)?

Answer: 54

Each step of 1 in x changes f(x) by 9, so the rule is f(x) $= 9x + 0$ .
$f(6) = 9(6) + 0 = 54$

Common mistakes

Reading f(3) as f multiplied by 3.
Substituting the output value into the rule when the question asked you to solve for the input.
Losing the sign when squaring a negative input: $(-3) 2$ is 9, not -9.
Computing f(g(x)) as g(f(x)) — the inner function is the one written next to the x.

What you should be able to do

Evaluate f(x) for a given input.
Solve f(x) $= k$ for the input that gives a required output.
Decide whether a relation is a function.
State the domain and range of a simple function.

Where this fits in the curriculum

Common Core

8.F.A.1
Grade 8 — Understand that a function assigns to each input exactly one output.
HSF-IF.A.1
High school — Understand the definition of a function and use domain and range correctly.
HSF-IF.A.2
High school — Use function notation, evaluate functions for inputs in their domains, and interpret statements that use it.