Precalculus-I · Unit 1 · Video 2 · Interactive Practice

Function Notation: Reading $f(x)$ and Functions in Tables

IKey Formulas

Formula	Name	What it says
$y = f(x)$	Function notation	$f$ is the rule, $x$ the input (independent variable), $y = f(x)$ the output (dependent variable)
$f(2005) = 300$	Reading a statement	Inside the parentheses: the input. Other side of $=$ : the output
$f(a + b) \ne f(a) + f(b)$	Expression as input	Work out $a + b$ first; that single value goes into $f$ (in general the two sides differ)
Each input $\to$ one output	Table test	Repeated outputs are allowed; one input with two different outputs is not

Key Insight: $f(x)$ is not $f$ times $x$ . The letter $f$ names a rule, and the parentheses hold what goes into it.

IIVisualization 1 — Rule, Input, Output

In every statement written with function notation, one symbol is the rule, one the input and one the output.

Statement

IIIVisualization 2 — An Expression as the Input

For the days-in-a-month function $D = f(m)$ , is $f(a + b)$ the same as $f(a) + f(b)$ ?

a 1 b 2

IVVisualization 3 — When Is a Table a Function?

A table represents a function exactly when no input value is paired with two different outputs.

Table

VQuiz Questions

Problem 1 · Interpret a Statement

Given: The function $T = g(d)$ gives the high temperature $T$ , in °F, in a city on day $d$ of March. What does $g(12) = 54$ mean?

✅ Correct! The input

12

is a day; the output

54

is a temperature.

❌ Input and output are swapped. The value inside the parentheses is the input, and here the input is the day

d

. March has no day 54.

❌ Nothing is multiplied.

g

is the name of a rule, and

g(12)

means "the rule

g

applied to

12

❌ Not quite.

g(12) = 54

pairs one input with one output; it says nothing about a change over several days.

Show solution

Compare $g(12) = 54$ with $T = g(d)$ .

Inside the parentheses: $d = 12$ , the input, which is a day of March.
On the other side of $=$ : $T = 54$ , the output, which is a temperature in °F.

So on March 12 the high temperature was 54 °F.

Problem 2 · Which Table Fails?

Given: Each option lists the (input, output) rows of a table. Which table does NOT represent a function?

✅ Correct! The input

4

is paired with both

-2

and

2

❌ A repeated output is allowed. Here an output appears more than once, but every input is different, so each input still has exactly one output.

❌ Not quite. The inputs

1

4

9

are all different, so each has a single output. Look for an input that appears twice with different outputs.

Show solution

Step 1: list the inputs of each table. Step 2: check whether any input is paired with two different outputs.

(a) inputs $1, 2, 3$ are different: function (output $4$ repeats, which is allowed).
(b) inputs $-2, 0, 2$ are different: function (output $0$ repeats, which is allowed).
(c) inputs $1, 4, 9$ are different: function.
(d) input $4$ appears twice, with outputs $-2$ and $2$ : not a function. Written in notation it would need $f(4) = -2$ and $f(4) = 2$ .

Problem 3 · Add First, or Apply First?

Given: A function $f$ is defined by the table

$x$	$1$	$2$	$3$	$4$	$5$	$6$
$f(x)$	$4$	$6$	$1$	$5$	$3$	$2$

What is $f(2 + 3)$ ?

What is $f(2) + f(3)$ ?

✅ Correct!

f(2 + 3) = 3

but

f(2) + f(3) = 7

: two different calculations, two different answers.

❌ That is $f(2) + f(3)$ . In

f(2 + 3)

the sum inside the parentheses is worked out first, and

f

is applied once, to

5

❌ $5$ is the input.

2 + 3 = 5

is what goes into

f

; the answer is the output in the column

x = 5

❌ Not quite. First

2 + 3 = 5

, then read the output under

x = 5

❌ That is $f(2 + 3)$ . Here

f

is applied to

2

and to

3

separately, and the two outputs are added.

❌ $2 + 3 = 5$ adds the inputs. Add the outputs instead:

f(2)

and

f(3)

❌ Not quite. Read

f(2)

and

f(3)

from the table, then add them.

Show solution

$f(2 + 3)$ : work out the input first, then apply $f$ once.

f(2 + 3) = f(5) = 3

$f(2) + f(3)$ : apply $f$ to each number, then add the outputs.

f(2) + f(3) = 6 + 1 = 7

Since $3 \ne 7$ , this table is an example where $f(a + b) \ne f(a) + f(b)$ .

Problem 4 · Completing a Table

Given: A table has the rows $(1, 7)$ , $(2, 7)$ , $(3, 8)$ , $(4, 9)$ and a fifth row $(2, m)$ . For which value of $m$ does the table represent a function?

✅ Correct! With

m = 7

the fifth row repeats the pair

(2, 7)

, so

2

still has only one output.

❌ Not quite. With

m = 8

the input

2

would be paired with both

7

and

8

❌ Repeated outputs are allowed. The output

7

already belongs to inputs

1

and

2

; that is fine. Any

m \ne 7

gives the input

2

a second, different output.

❌ An input may appear twice with the same output. The test fails only when one input is paired with two different outputs.

Show solution

The input $2$ already appears in the row $(2, 7)$ , so the table already says $f(2) = 7$ .

If $m \ne 7$ , the table would need $f(2) = 7$ and $f(2) = m$ , two different values for one input: not a function.
If $m = 7$ , the new row is the same pair $(2, 7)$ again; every input is still paired with exactly one output.

So $m = 7$ , and the function is $f(1) = 7$ , $f(2) = 7$ , $f(3) = 8$ , $f(4) = 9$ .

Solved: 0 / 4

Function Notation: Reading f(x)f(x)f(x) and Functions in Tables

IKey Formulas

IIVisualization 1 — Rule, Input, Output

IIIVisualization 2 — An Expression as the Input

IVVisualization 3 — When Is a Table a Function?

VQuiz Questions

Function Notation: Reading $f(x)$ and Functions in Tables