1.2 Functions and Notation — Transformations

Standards: G-CO.A.2

This course is four units long and every one of them runs on the same idea: a rule that takes something in and gives something out.

You already use rules like that. A pay rate is a rule — hours in, wages out. A recipe scaled for six people is a rule. A thermostat is a rule.

Geometry writes those rules down in a particular notation, and once you can read the notation the rest of the course is mostly arithmetic you can already do.

Before you read on: if 12 hours of work pays $186, what is the rule?

Given a named rule and an input, you will replace every occurrence of the variable, compute the output, and explain what the parentheses contain.

Definitions and rules

A function is a named rule

f(x) = x + 4

f is the name. x is the input. x + 4 is the rule.

Why it has to be this: the parentheses hold what goes in, not a multiplication.

Why the notation is worth the trouble

Writing “add four to a number” in words is fine for one rule. It stops working when a rule has two inputs, or when you need to apply one rule and then another.

Notation lets you write f(g(x)) and be understood exactly. That is the whole reason for it.

Worked examples

Example 1 — a rule with two steps

If g(x) = 2x + 1, find g(3).

Suggested work

Givens

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

Tools

substitute, then work left to right

Primitives

x = 3

Computation

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

Notice: the whole rule applies, not just the first part of it.

Example 2 — a negative input

If h(x) = x + 9, find h(−4).

Suggested work

Givens

h(x) = x + 9 x = −4

Tools

substitute the input for x

Primitives

x = −4

Computation

h(−4) = (−4) + 9 = 5

Notice: brackets around the input keep the sign attached to it.

Example 3 — putting a number in

If f(x) = x + 4, find f(10).

Suggested work

Givens

f(x) = x + 4 x = 10

Tools

substitute the input for x

Primitives

x = 10

Computation

f(10) = (10) + 4 = 14

Notice: the input replaces every x. Nothing else changes.

Common traps

The mistake that costs the most marks

f(10) does not mean f times 10. The parentheses hold the input. If f(x) = x + 4 then f(10) = 14, not 40. Every rule in this course uses this notation, so it is worth being certain now.

Practice

If f(x) = x + 7, find f(5).
Answer — tap to show
12
Suggested work — tap to show
Givens
f(x) = x + 7 x = 5
Tools
substitute for x
Primitives
x = 5
Computation
f(5) = (5) + 7 = 12
If p(x) = x + 6, find p(−9).
Answer — tap to show
−3
Suggested work — tap to show
Givens
p(x) = x + 6 x = −9
Tools
substitute for x
Primitives
x = −9
Computation
p(−9) = (−9) + 6 = −3
A classmate uses m(x) = x + x + 5 and writes m(3) = 3 + x + 5 = 8. Diagnose the error and give the correct value.
Answer — tap to show
They replaced only one occurrence of x. Every x takes the input 3, so m(3) = 3 + 3 + 5 = 11.
Suggested work — tap to show
Givens
m(x) = x + x + 5 input: 3 classmate wrote m(3) = 8
Tools
Replace every x with the input.
Primitives
x = 3 in both locations
Computation
m(3) = (3) + (3) + 5 = 11
A classmate is told f(x) = x + 4 and writes f(10) = 40. Find their mistake.
Answer — tap to show
They read f(10) as f times 10. The correct value is 14.
Suggested work — tap to show
Givens
f(x) = x + 4 Classmate wrote f(10) = 40
Tools
substitute the input for x
Primitives
x = 10
Computation
f(10) = (10) + 4 = 14 The mistake: multiplying by the input instead of substituting it.
If q(x) = 3x + 2, find q(4).
Answer — tap to show
14
Suggested work — tap to show
Givens
q(x) = 3x + 2 x = 4
Tools
substitute, then work left to right
Primitives
x = 4
Computation
q(4) = 3(4) + 2 = 12 + 2 = 14