MCF3M Functions & Applications · Unit 1 — Introduction to the Quadratic Function

Lesson 1.3

Working with Function Notation

f(x) is a compact way to say “the output for this input”. Learn to read it, evaluate it, link it to tables, graphs and points, and explain it in real situations.

Estimated time: 30–45 minutes

Lesson progress0%

Learning goals

interpret f(x) — and know it is not multiplication
evaluate a function at positive, negative and simple algebraic inputs
read function notation from tables and graphs in both directions
connect f(a) = b to the point (a, b)
interpret function notation in context, with units

Step 1 of 4

Warm Up

Real-world idea

Renting a bike

A bike rental charges a $5 helmet fee plus $8 for each hour.

Saying “the cost when the rental lasts 3 hours” over and over gets tiring. Is there a shorter way to write it?

Warm Up

A bike rental charges a $5 helmet fee plus $8 per hour.

What is the total cost for 3 hours?

Your answer ($)

Mathematicians write C(3) = 29: “the cost for 3 hours is $29”. That short form is function notation.

Step 2 of 4

Learn

Part 1 — What f(x) means

f is the name of the function. x is the input. f(x) is the output produced by that input.

Input

Function f

3x² − 2

Output

f(2) = 10: put 2 in, get 10 out.

f(x) is NOT f × x

The brackets hold the input. f(2) means “the output when the input is 2”, never “f times 2”.

Try it

What does g(7) mean?

ag multiplied by 7bThe output of function g when the input is 7cThe input of g when the output is 7d7 copies of g

Part 2 — Numerical inputs

Use f(x) = 3x² − 2. Replace every x with the input, in brackets.

Positive input: f(4)

f(4) = 3(4)² − 2
= 3(16) − 2square first
= 46

Negative input: f(−2)

f(−2) = 3(−2)² − 2brackets around −2
= 3(4) − 2(−2)² = 4
= 10

Always bracket negatives

(−2)² = 4, but −2² = −4. Without brackets you'd get 3(−4) − 2 = −14 — wrong.

Try it

f(x) = 3x² − 2. Find f(−3).

Your answer

Part 3 — Algebraic inputs

The input can be a letter or an expression. Everything in the brackets replaces x.

f(x) = 3x² − 2

f(a) = 3a² − 2replace x with a
f(2m) = 3(2m)² − 2 = 3(4m²) − 2 = 12m² − 2(2m)² = 4m²
f(n + 1) = 3(n + 1)² − 2substitute the whole input
= 3(n² + 2n + 1) − 2 = 3n² + 6n + 1

Everything inside the brackets replaces every x.

Try it

f(x) = 3x² − 2. Which is f(2m)?

a6m² − 2b12m² − 2c6m − 2d3(2m) − 2

Part 4 — Tables

A table can be read in two directions.

x	f(x)
-1	6
0	3
1	2
2	3
3	6
4	11

“What is f(1)?” — the input is given. Find x = 1, read the output: f(1) = 2.

“For what x is f(x) = 6?” — the output is given. Find 6 in the f(x) row: x = −1 or x = 3. There can be more than one input.

Try it

x: −1, 0, 1, 2, 3, 4 f(x): 6, 3, 2, 3, 6, 11

For what value of x is f(x) = 11?

Your answer

Part 5 — Graphs and points

Every statement f(a) = b is a point (a, b) on the graph. The graph below is f(x) = x².

f(x) = x²

f(x) = x²

If f(3) = 7, then (3, 7) is on the graph. If (5, 12) is on the graph, then f(5) = 12.

Try it

The point (5, 12) is on the graph of h. Which statement is true?

ah(12) = 5bh(5) = 12ch = 5 × 12dh(5, 12) = 0

Real-world idea

Part 6 — The garden fountain

A garden fountain shoots an arc of water. h(d) = −0.5d² + 2d + 1.6 models the water's height h, in metres, at a horizontal distance d metres from the nozzle.

h(2) = −0.5(4) + 4 + 1.6 = 3.6. In words, with units: at 2 m from the nozzle, the water is 3.6 m high.

Working backwards: if f(x) = 4x − 3 and f(x) = 17, solve 4x − 3 = 17 → x = 5. You're finding the input that gives that output.

Try it

h(d) = −0.5d² + 2d + 1.6 gives the height (m) of a fountain's water stream at horizontal distance d (m) from the nozzle.

What does h(2) = 3.6 mean?

aThe water is 2 m high, 3.6 m from the nozzle.bAt 2 m horizontally from the nozzle, the water is 3.6 m high.cThe water travels 3.6 m in 2 seconds.dh times 2 is 3.6.

Working with Function Notation

Warm Up

Renting a bike

Learn

Part 1 — What f(x) means

Part 2 — Numerical inputs

Part 3 — Algebraic inputs

Part 4 — Tables

Part 5 — Graphs and points

Part 6 — The garden fountain

Practice

Build it

Apply it

Challenge

Explain it

Mastery Check

Lesson 1.3 mastery check