3. Function Notation

What Is Function Notation?

A function describes a relationship between an:

input

and an:

output.

You may already be familiar with equations such as:

y = 2x + 3

Function notation gives us another way to write the same relationship:

f(x) = 2x + 3

These two equations can describe the same function.

However, function notation gives us a clearer way to identify and work with:

specific functions and their inputs.

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Understanding f(x)

The expression:

f(x)

is read as:

"f of x".

It does not mean:

f × x.

The letter f is the:

name of the function.

The value inside the parentheses is the:

input.

So:

f(x)

means:

the output of function f when the input is x.

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Function Notation as a Machine

One useful way to think about a function is as a:

machine.

An input enters the machine.

The function performs a rule.

An output comes out.

For example:

f(x) = 2x + 3

If the input is:

4

then:

f(4) = 2(4) + 3

f(4) = 8 + 3

f(4) = 11

So the function takes:

4 → 11

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The Parts of Function Notation

Consider:

f(x) = 3x - 5

There are several important parts.

f = name of the function

x = input variable

3x - 5 = function rule

f(x) = output produced by the function

The equation tells us:

Take the input, multiply it by 3, then subtract 5.

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Function Names

Functions are often named using letters such as:

f, g, h, p, or C.

For example:

f(x) = 2x + 1

g(x) = x²

h(x) = √x

The letters simply allow us to:

identify different functions.

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Why Do We Need Function Names?

Suppose we have two relationships:

y = 2x + 1

and:

y = x²

Writing both using y can become confusing.

Instead, we can write:

f(x) = 2x + 1

and:

g(x) = x²

Now we can clearly refer to:

function f

and:

function g.

This is one major advantage of:

function notation.

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Evaluating a Function

To evaluate a function means to find its output for a particular:

input.

Suppose:

f(x) = 2x + 5

Find:

f(3)

The notation f(3) means:

use 3 as the input.

Replace every x with:

3.

Therefore:

f(3) = 2(3) + 5

f(3) = 6 + 5

f(3) = 11

So:

f(3) = 11

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The Substitution Method

When evaluating a function, use these steps:

Step 1: Identify the input.

Step 2: Substitute the input for every x in the function.

Step 3: Calculate carefully.

Step 4: Write the answer using function notation.

For example:

f(x) = 4x - 7

Find:

f(5)

Substitute:

f(5) = 4(5) - 7

Calculate:

f(5) = 20 - 7

Therefore:

f(5) = 13

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Evaluating at Zero

Suppose:

f(x) = 3x + 8

Find:

f(0)

Substitute 0 for x:

f(0) = 3(0) + 8

f(0) = 8

Therefore:

f(0) = 8

Notice that f(0) does not mean the answer must be:

zero.

It means:

find the output when the input is zero.

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Evaluating Negative Inputs

Suppose:

f(x) = 2x + 7

Find:

f(-3)

Substitute:

f(-3) = 2(-3) + 7

f(-3) = -6 + 7

Therefore:

f(-3) = 1

Negative inputs are handled in exactly the same way as:

positive inputs.

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Be Careful with Negative Numbers

Suppose:

f(x) = x² + 2

Find:

f(-4).

Substitute:

f(-4) = (-4)² + 2

f(-4) = 16 + 2

f(-4) = 18

The parentheses are important.

(-4)² = 16

This is why it is a good habit to place substituted values inside:

parentheses.

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Evaluating Quadratic Functions

Consider:

f(x) = x² - 3x + 2

Find:

f(4).

Substitute 4 for every x:

f(4) = (4)² - 3(4) + 2

f(4) = 16 - 12 + 2

Therefore:

f(4) = 6

Every occurrence of x must be:

replaced.

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Evaluating a Fraction

Suppose:

f(x) = (x + 4) / 2

Find:

f(6).

Substitute:

f(6) = (6 + 4) / 2

f(6) = 10 / 2

Therefore:

f(6) = 5

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Evaluating a Square Root Function

Suppose:

f(x) = √(x + 1)

Find:

f(8).

Substitute:

f(8) = √(8 + 1)

f(8) = √9

Therefore:

f(8) = 3

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Function Notation and Tables

Function notation can also describe information in a:

table.

Consider:

From the table:

f(-2) = 1

f(0) = 5

f(2) = 9

The x-column contains the:

inputs.

The f(x)-column contains the:

outputs.

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Reading Function Notation from a Table

Using the same table:

What is:

f(1)?

Find:

x = 1

Then read the corresponding output.

Therefore:

f(1) = 7

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Function Notation and Graphs

A graph of:

y = f(x)

shows the outputs of a function for different:

x-values.

Suppose a graph contains the point:

(3, 7).

This means:

when x = 3, the output is 7.

Using function notation:

f(3) = 7

So the coordinate:

(3, 7)

can be interpreted as:

input 3 → output 7.

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Connecting Coordinates and Function Notation

If:

f(2) = 5

then the graph of f contains the point:

(2, 5).

If:

f(-3) = 4

then the graph contains:

(-3, 4).

If:

f(0) = -2

then the graph contains:

(0, -2).

This connection is extremely useful:

f(a) = b ↔ (a, b)

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Writing Equations in Function Notation

Suppose:

y = 5x - 2

To write this using function notation, replace y with:

f(x).

Therefore:

f(x) = 5x - 2

The mathematical relationship has not changed.

We have simply given the function:

a name.

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Another Example

Rewrite:

y = x² + 4x - 1

using function notation.

Replace y with f(x):

f(x) = x² + 4x - 1

Now the function can easily be referred to as:

function f.

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Different Function Names

There is nothing special about the letter:

f.

We could write:

g(x) = 2x + 7

or:

h(x) = x² - 3

or:

p(x) = 4x

The function name can often be chosen to make the situation easier to:

understand.

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Meaningful Function Names

In real-world applications, letters may represent:

quantities.

For example:

C = 12n

could represent the cost C of buying n tickets at $12 each.

Then:

C(5)

means:

the cost of buying 5 tickets.

Calculate:

C(5) = 12(5)

C(5) = 60

So:

5 tickets cost $60.

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Function Notation in Context

Suppose:

T = 20 - 2h

represents the temperature T after h hours.

What does:

T(3)

mean?

It means:

the temperature after 3 hours.

Calculate:

T(3) = 20 - 2(3)

T(3) = 14

Therefore:

the temperature after 3 hours is 14 degrees.

Function notation helps communicate the:

meaning of the variables.

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Input and Output

Consider:

P(t) = 100 + 20t

Here:

t

is the input.

P(t)

is the output.

If P represents a population and t represents time, then:

P(5)

means:

the population at time 5.

This is more informative than simply writing:

y.

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Function Notation Is Not Multiplication

This is one of the most important ideas in this topic.

f(x)

does not mean:

f × x.

The parentheses indicate the:

input to the function.

Compare:

3(x)

which means multiplication,

with:

f(x)

which means the output of function f for input x.

The notation may look similar, but the meanings are:

different.

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The Value Inside the Parentheses

The value inside the parentheses tells us:

what to substitute for the input variable.

If:

f(x) = x² + 1

then:

f(3) means substitute 3.

f(-2) means substitute -2.

f(10) means substitute 10.

But the input does not have to be:

a number.

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Evaluating with Another Variable

Suppose:

f(x) = 2x + 5

Find:

f(a).

Replace x with a:

f(a) = 2a + 5

Nothing else can be simplified because a is:

unknown.

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Evaluating an Expression

Suppose:

f(x) = 3x - 2

Find:

f(n + 1).

Replace every x with:

n + 1.

Therefore:

f(n + 1) = 3(n + 1) - 2

Expand:

f(n + 1) = 3n + 3 - 2

Therefore:

f(n + 1) = 3n + 1

This shows that function inputs can be:

expressions as well as numbers.

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Another Expression Example

Suppose:

f(x) = x² + 4

Find:

f(2a).

Substitute:

f(2a) = (2a)² + 4

Therefore:

f(2a) = 4a² + 4

Again, the entire input replaces:

x.

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Comparing f(x) and f(2)

Suppose:

f(x) = 3x + 1

Then:

f(x)

describes the general output for any allowed:

x-value.

But:

f(2)

asks for one specific output.

Calculate:

f(2) = 3(2) + 1

f(2) = 7

So:

f(x) is a general expression.

f(2) is a specific value.

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Comparing x and f(x)

The symbols:

x

and:

f(x)

do not represent the same thing.

x represents:

the input.

f(x) represents:

the output produced from that input.

For:

f(x) = 2x + 3

if:

x = 4

then:

f(x) = 11.

Input:

4

Output:

11.

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More Than One Function

Suppose:

f(x) = 2x + 1

and:

g(x) = x²

Find:

f(3).

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

Now find:

g(3).

g(3) = 3² = 9

The same input can produce different outputs because:

the functions have different rules.

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Why Function Names Are Useful

Suppose we want to compare:

f(x) = 2x + 1

g(x) = x²

h(x) = 10 - x

Function notation allows us to discuss all three relationships:

at the same time.

We can ask:

f(4)

g(4)

h(4)

without confusing which rule should be:

used.

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Functions Can Be Compared

Using:

f(x) = 2x + 1

and:

g(x) = x²

at x = 3:

f(3) = 7

g(3) = 9

Therefore:

g(3) > f(3).

Function notation makes comparisons between functions:

clear and concise.

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Finding the Input from an Output

Sometimes we know the output and must determine:

the input.

Suppose:

f(x) = 3x + 2

and:

f(x) = 17.

This means:

3x + 2 = 17

Subtract 2:

3x = 15

Divide by 3:

x = 5

Therefore:

f(5) = 17.

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Another Reverse Example

Suppose:

g(x) = 2x - 6

Find x if:

g(x) = 10.

Write:

2x - 6 = 10

Add 6:

2x = 16

Divide by 2:

x = 8

Therefore:

g(8) = 10.

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Function Notation and Domain

Function notation also connects naturally with:

domain and range.

The values that may be placed inside the parentheses belong to the:

domain.

The resulting values of the function belong to the:

range.

For example:

f(3) = 11

means:

3 is an input

and:

11 is the corresponding output.

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Restricted Inputs

Consider:

f(x) = 1 / x

The expression:

f(0)

would mean:

1 / 0

which is undefined.

Therefore:

f(0) is undefined.

This occurs because 0 is not part of the:

domain.

Function notation does not remove the restrictions of a:

function.

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Function Notation and Graphs

Consider:

f(x) = x²

The graph represents every pair:

(x, f(x)).

For example:

f(-2) = 4

f(-1) = 1

f(0) = 0

f(1) = 1

f(2) = 4

These correspond to the points:

(-2, 4)

(-1, 1)

(0, 0)

(1, 1)

(2, 4)

Together, these points form the familiar:

parabola.

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Worked Example 1

Given:

f(x) = 5x - 4

find:

f(6).

Substitute:

f(6) = 5(6) - 4

f(6) = 30 - 4

Therefore:

f(6) = 26

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Worked Example 2

Given:

g(x) = x² + 3x

find:

g(-2).

Substitute:

g(-2) = (-2)² + 3(-2)

g(-2) = 4 - 6

Therefore:

g(-2) = -2

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Worked Example 3

Given:

h(x) = 4 - 2x

find:

h(0).

Substitute:

h(0) = 4 - 2(0)

Therefore:

h(0) = 4

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Worked Example 4

Given:

p(x) = x² - 5

find:

p(a).

Replace x with a:

p(a) = a² - 5

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Worked Example 5

Given:

f(x) = 2x + 3

find:

f(x + 1).

Replace every x in the rule with:

x + 1.

Therefore:

f(x + 1) = 2(x + 1) + 3

Expand:

f(x + 1) = 2x + 2 + 3

Therefore:

f(x + 1) = 2x + 5

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Function Notation from a Word Rule

Suppose a function:

multiplies an input by 4 and then subtracts 3.

We can write:

f(x) = 4x - 3

This converts a verbal rule into:

standard function notation.

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Another Word Rule

Suppose a function:

squares the input and then adds 7.

We can write:

g(x) = x² + 7

The name g simply identifies:

this particular function.

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From a Table to Function Notation

Consider:

The outputs increase by:

2

for every increase of 1 in x.

The function rule is:

f(x) = 2x + 3

We can verify:

f(0) = 3

f(1) = 5

f(2) = 7

and so on.

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Function Notation in Real Life

Function notation is useful whenever one quantity depends on:

another quantity.

For example:

C = cost of n items

d(t) = distance travelled after time t

T(t) = temperature at time t

P(t) = population at time t

A(r) = area of a circle with radius r

The notation communicates both:

the quantity being calculated and the variable it depends on.

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Example: Area of a Circle

The area of a circle depends on its:

radius.

Instead of simply writing:

A = πr²

we can write:

A(r) = πr²

This emphasizes that area is a:

function of radius.

For a radius of 5:

A(5) = π(5)²

A(5) = 25π

So A(5) means:

the area of a circle with radius 5.

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Example: Distance Travelled

Suppose a vehicle travels at a constant speed of:

60 km/h.

Its distance after t hours can be written:

d(t) = 60t

Then:

d(2) = 120

means:

the distance travelled after 2 hours is 120 km.

The notation tells us exactly what the:

input and output represent.

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Why Function Notation Is Better Than Always Using y

Writing:

y = 60x

tells us the mathematical relationship.

But:

d(t) = 60t

provides additional meaning.

It tells us:

d represents distance

and:

t represents time.

Function notation can therefore make mathematical models:

more informative.

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Advantages of Function Notation

Function notation provides several important advantages.

It:

• clearly identifies the function
• clearly identifies the input
• makes evaluating functions easy to express
• allows several functions to be discussed at once
• connects equations, tables, and graphs
• makes real-world models easier to interpret
• allows functions to be combined and compared
• provides a standard language used throughout higher mathematics

Function notation becomes increasingly important in:

algebra, calculus, statistics, science, economics, and engineering.

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Common Mistake: Treating f(x) as Multiplication

Incorrect idea:

f(x) = f × x

Correct idea:

f(x) means the output of function f for input x.

Always read:

f(x)

as:

"f of x".

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Common Mistake: Forgetting to Substitute Every x

Suppose:

f(x) = x² + 3x - 1

and we want:

f(2).

Incorrect:

2² + 3x - 1

The x in 3x was not replaced.

Correct:

f(2) = (2)² + 3(2) - 1

f(2) = 4 + 6 - 1

f(2) = 9

Every x must be:

replaced by the input.

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Common Mistake: Losing Parentheses

Suppose:

f(x) = x² - 4x

and we want:

f(-3).

Write:

f(-3) = (-3)² - 4(-3)

not:

-3² - 4 × -3

Using parentheses helps prevent:

sign errors.

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Common Mistake: Confusing Input and Output

If:

f(4) = 11

then:

4 is the input

and:

11 is the output.

It does not mean that:

f = 4

or that:

x = 11.

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Common Mistake: Thinking f Is Always the Function Name

The function does not have to be called:

f.

For example:

g(x) = 3x

h(t) = t²

C = 5n + 10

are all valid examples of:

function notation.

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A Useful Function Notation Strategy

When you see something such as:

f(7)

think:

"Put 7 into function f."

Then:

1. Find the rule for f.
2. Replace every input variable with 7.
3. Calculate the result.
4. Write the output.

For example:

f(x) = 3x² - 2

Find:

f(2).

Substitute:

f(2) = 3(2)² - 2

f(2) = 12 - 2

f(2) = 10

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Check Your Understanding

1. What does f(x) mean?

2. How do you read f(x) aloud?

3. Does f(x) mean f multiplied by x? Explain.

4. In f(x) = 3x + 2, what is the input variable?

5. In f(x) = 3x + 2, what does f represent?

6. If f(x) = 2x + 5, find f(4).

7. If f(x) = 7x - 3, find f(0).

8. If f(x) = 4x + 1, find f(-2).

9. If g(x) = x² + 2, find g(3).

10. If g(x) = x² - 5x, find g(-2).

11. If h(x) = √(x + 4), find h(5).

12. Rewrite y = 6x - 7 using function notation.

13. Rewrite y = x² + 4x + 1 using function notation.

14. If f(3) = 8, what point lies on the graph of f?

15. If the point (-2, 7) lies on the graph of g, write this using function notation.

16. Given f(x) = 5x + 1, find f(a).

17. Given f(x) = 2x - 3, find f(n + 1).

18. Given g(x) = x² + 4, find g(2a).

19. If f(x) = 3x + 2 and f(x) = 20, find x.

20. If g(x) = 5x - 4 and g(x) = 31, find x.

21. Explain the difference between x and f(x).

22. Explain the difference between f(x) and f(5).

23. Give one advantage of naming functions f, g, and h instead of writing every function using y.

24. Write a function that doubles an input and then adds 7.

25. A taxi charges a starting fee of $4 plus $3 per kilometre. Write the total cost C as a function of distance d.

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Key Terms

• Function: Relationship that assigns each allowed input exactly one output.
• Function notation: A system for representing functions using expressions such as f(x).
• f(x): Output of function f for the input x.
• Function name: Symbol used to identify a function, such as f, g, or h.
• Input: Value supplied to a function.
• Output: Value produced by a function.
• Variable: Symbol representing a value that can change.
• Independent variable: Variable used as the input.
• Dependent variable: Variable whose value depends on the input.
• Evaluate: Calculate the output of a function for a given input.
• Substitute: Replace a variable with a specified value or expression.
• Function rule: Mathematical operation or expression connecting inputs to outputs.
• Ordered pair: Pair of values written as (x, y).
• Domain: Set of allowed inputs.
• Range: Set of possible outputs.
• Mathematical model: Function used to represent a real-world relationship.

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Key Takeaways

• Function notation provides a standard way to represent relationships between inputs and outputs.
• f(x) is read as "f of x."
• f(x) does not mean f multiplied by x.
• The letter f names the function.
• The value inside the parentheses identifies the input.
• f(x) represents the output of the function.
• To evaluate a function, substitute the given input for every occurrence of the input variable.
• Using parentheses during substitution helps prevent sign and exponent errors.
• f(3) means the output of function f when the input is 3.
• If f(3) = 7, then the point (3, 7) lies on the graph of f.
• Functions can be named using letters other than f.
• Function notation allows several different functions to be discussed clearly at the same time.
• A function can be evaluated using numerical inputs, negative inputs, variables, or algebraic expressions.
• Function notation can be read from equations, tables, graphs, and real-world models.
• Domain values are the possible values that can appear as inputs.
• Range values are the possible resulting outputs.
• Real-world function names such as C, d(t), and A(r) can communicate what the quantities represent.
• Function notation is more descriptive than always using x and y.
• Function notation becomes increasingly important in advanced algebra, calculus, science, and other quantitative subjects.
• The central idea is simple: input → function rule → output.