MCR3U Functions · Unit 1 — Introduction to Functions

Lesson 1.2

Function Notation

f(3) looks like multiplication, but it isn't. Learn the shorthand mathematicians use for “put this input in, get this output out”.

Estimated time: 30–45 minutes

Lesson progress0%

Learning goals

explain what function notation means
evaluate a function at positive and negative numerical inputs
evaluate a function at algebraic inputs such as f(a), f(3b) and f(c + 2)
evaluate expressions such as 2f(3), f(2) + f(5) and f(4) + g(2)
read function notation from a graph and from a table
find the input when the output is given
interpret function notation in a real-world situation

Step 1 of 4

Warm Up

One quick question, then a shorter way to write the same idea.

You already know how to put a number into a rule and get one number out. Let's check that first.

Warm-up question

A rule connects two numbers: y = 2x + 3.

What is y when x = 4?

Your answer

Now the same idea, written differently

The rule y = 2x + 3 can also be written as f(x) = 2x + 3. It is the exact same relationship — just a shorter way of naming it.

Because putting 4 in gives 11 out, we write f(4) = 11.

The same work, in function notation

f(x) = 2x + 3f is the name of the function.
f(4) = 2(4) + 34 is the input.
f(4) = 1111 is the output.

Three parts, one line

In f(4) = 11: f is the function's name, 4 is the input, and 11 is the output.

Step 2 of 4

Learn

What f(x) means, and every place you will meet it.

What does f(x) mean?

f(x) means the output of the function f when the input is x. Think of the function as a machine: a number goes in, the rule works on it, and one number comes out.

Input

Function f

2x + 3

Output

Input 4, rule 2x + 3, output 11 — written as f(4) = 11.

Key term

Function notation

A way of writing a function so the name, the input and the output are all visible at once.

f(4) = 11 is read “f of 4 equals 11”.

The most common mistake

f(x) does NOT mean f × x. The brackets are not multiplication here — they hold the input. f(4) is a single value: the output you get when 4 goes in.

Try it

A function is named h.

What does h(5) mean?

ah multiplied by 5bThe output of h when the input is 5cThe input of h when the output is 5d5 added to h

Putting numbers in

To evaluate a function, replace every x with the input and simplify. Take f(x) = x² − 4x + 7 and find f(3).

f(3) step by step

f(x) = x² − 4x + 7Start with the rule.
f(3) = (3)² − 4(3) + 7Replace every x with 3.
= 9 − 12 + 7Square, then multiply.
= 4So f(3) = 4.

Negative inputs

Negative inputs work the same way — but the brackets matter. Take g(x) = 2x² + 3x − 1 and find g(−2).

Worked example

g(x) = 2x² + 3x − 1. Find g(−2).

Step 1
Put the input in brackets.
g(−2) = 2(−2)² + 3(−2) − 1
Step 2
Square the bracket first.
= 2(4) − 6 − 1
Step 3
Multiply, then simplify.
= 8 − 6 − 1 = 1

Answer: g(−2) = 1

Always use brackets

(−2)² = 4, but −2² = −4. Writing the brackets protects you from a sign error.

Try it

f(x) = x² − 4x + 7

Find f(5).

Your answer

Try it

g(x) = 2x² + 3x − 1

Find g(−3).

Your answer

The input does not have to be a number

The input can be a variable, or a whole expression. The rule does not change — whatever sits inside the brackets replaces every x.

Use f(x) = 2x² − 3x + 1 for the next two examples.

Example 1 — f(a)

f(a) = 2(a)² − 3(a) + 1Every x becomes a.
f(a) = 2a² − 3a + 1Nothing else changes.

Example 2 — f(3b)

f(3b) = 2(3b)² − 3(3b) + 1Every x becomes 3b.
(3b)² = 9b²The 3 gets squared too.
f(3b) = 18b² − 9b + 12 × 9b² = 18b².

Square the whole input

(3b)² = 9b², not 3b². The bracket means the 3 and the b are both squared.

Worked example

g(x) = x² + 4x − 1. Find g(c + 2).

Step 1
Substitute before you expand. The whole expression c + 2 replaces every x.
g(c + 2) = (c + 2)² + 4(c + 2) − 1
Step 2
Expand the square and distribute the 4.
= c² + 4c + 4 + 4c + 8 − 1
Step 3
Collect like terms.
= c² + 8c + 11

Answer: g(c + 2) = c² + 8c + 11

Everything inside the brackets replaces every x.

If the input is c + 2, then the whole expression c + 2 goes everywhere x used to be.

Try it

f(x) = 2x² − 3x + 1

Which expression is f(4b) in simplified form?

a32b² − 12b + 1b8b² − 12b + 1c32b² − 3b + 1d2b² − 3b + 1

Expressions built from function values

Sometimes you need to combine outputs. Work out each function value first, then do the arithmetic.

For f(x) = x² − 2x, find f(2) + f(5).

f(2) + f(5)

f(2) = (2)² − 2(2) = 0First output.
f(5) = (5)² − 2(5) = 15Second output.
f(2) + f(5) = 0 + 15 = 15Add the two outputs.

3f(2) is not f(6)

A number outside the brackets multiplies the output: 3f(2) means 3 × f(2). A number inside the brackets is the input. Changing 2 to 6 asks a completely different question.

Two different functions

You may also see two functions in the same expression. For f(x) = 2x + 1 and g(x) = x², find f(3) + g(3).

f(3) + g(3)

f(3) = 2(3) + 1 = 7Use the rule for f.
g(3) = (3)² = 9Use the rule for g.
f(3) + g(3) = 7 + 9 = 16Add the two outputs.

Try it

f(x) = 2x + 1 and g(x) = x²

Find the value of 2f(4) − g(2).

Your answer

Reading function notation from a graph

A graph shows every input-output pair at once. On the graph of f(x) = x² below, the marked points show f(2) = 4 and f(−2) = 4.

To find f(2): start at x = 2, move straight up to the curve, then read the height. That height is f(2).

The graph of f(x) = x², with two outputs marked.

f(x) = x²

Going the other way

f(x) = 4 asks the opposite question: the output is 4, so which inputs produce it? Start at 4 on the vertical axis and look across at the curve.

Here the curve reaches a height of 4 twice, so x = 2 and x = −2 both work.

Back to Lesson 1.1

A function gives only one output for each input — but two different inputs are allowed to share the same output. That is exactly what you see here.

Try it

Use the graph of f(x) = x² shown above.

If f(x) = 9, what are all the possible inputs?

ax = 3 onlybx = 3 or x = −3cx = 81dx = 4.5

Reading function notation from a table

A table works the same way. Find the input in the x column, then read across to the f(x) column.

A table of values for f
x	f(x)
-2	5
-1	2
0	1
1	2
2	5

From the table, f(−2) = 5: the input −2 gives the output 5.

But f(x) = 2 asks the reverse question. Look down the f(x) column for 2 — it appears twice, so x = −1 or x = 1.

f(−2) = ? gives the input and asks for the output. f(x) = 2 gives the output and asks for the input.

Reading which one is being asked is half the work.

Try it

Use the table of values above.

What is f(2)?

Your answer

Working backwards with algebra

You can answer the same backwards question from an equation. For f(x) = 3x + 4, suppose f(x) = 19. That means the output is 19, and you need the input.

Finding the input

3x + 4 = 19Set the rule equal to the known output.
3x = 15Subtract 4 from both sides.
x = 5Divide both sides by 3.

Same thinking, three places

On a graph you look across from the output. In a table you scan the output column. With an equation you solve for x. All three answer the same question.

Try it

f(x) = 3x + 4

If f(x) = 25, what is x?

Your answer

Real-world idea

Function notation in real life

A ride costs a $4 starting fee plus $2.50 for every kilometre. Written as a function: C(d) = 4 + 2.5d, where d is the distance in kilometres and C(d) is the cost in dollars.

Before calculating anything, read C(6). It means: the cost of travelling 6 km.

C(6)

C(6) = 4 + 2.5(6)The input 6 is a distance.
C(6) = 4 + 15Multiply.
C(6) = 19A 6 km ride costs $19.

The statement C(d) = 29 says the opposite: the cost is $29, and the question is how far you travelled.

Try it

A ride costs a $4 starting fee plus $2.50 per kilometre, so C(d) = 4 + 2.5d.

What is C(8), in dollars? (Enter the number only.)

Your answer

Curriculum connection

These are the words your teacher and the Ontario curriculum use. Learn them here and use them when you explain your thinking.

Curriculum language · A1.2Substitute into and evaluate functions using function notation: What this meansPut the input into the rule — a number, a variable or a whole expression — and simplify to find the output.

Step 3 of 4

Practice

Build the skill, then use it. Hints are there when you want them.

Function Notation

Warm Up

Now the same idea, written differently

Learn

What does f(x) mean?

Function notation

Putting numbers in

Negative inputs

The input does not have to be a number

Expressions built from function values

Two different functions

Reading function notation from a graph

Going the other way

Reading function notation from a table

Working backwards with algebra

Function notation in real life

Practice

Build it

Apply it

Challenge

Explain it

Mastery Check

Lesson 1.2 mastery check