Functions and relationships (Grade 9) — Mathematics Grades 8–11

Working out outputs and inputs for a rule, and writing the rule that links input and output.

Gr 9 Terms 1 and 3 — Functions and relationshipsTerm 1 and Term 3

Practise Functions and relationships in the app →

What gets asked

Find the output of a flow diagram or formula for a given input.
Work backwards to find the input that gives a stated output.
Write a formula linking input x and output y.
Determine the output of an equation for a stated x-value.

You must be able to

Apply a flow diagram's operations to an input, in the order shown.
Undo a flow diagram's operations in reverse order to find an input.
Turn a description or table into a formula for y in terms of x.
Substitute a given x-value into an equation to find y.

Traps that cost marks

Applying a flow diagram’s operations in the order the words are written, not the order shown. Follow the diagram left to right exactly as drawn, even if the words come in a different order.
Using the forward operations again instead of working backwards. To find an input, undo each operation with its opposite, in reverse order.
Losing the sign when a negative input is squared. Put a negative value in brackets before squaring it: $(- 3)^{2} = 9$ .

Worked example

A flow diagram multiplies an input x by 4, then subtracts 3, to give output y. Find y when x = -2.

Multiply first: $4 \times (- 2) = - 8$ .
Then subtract 3: $- 8 - 3 = - 11$ .
The output is $y = - 11$ .

Input and output values

Following a rule forwards to get an answer, and backwards to find where you started.

input: The number you put into a rule.
output: The number the rule hands back to you.
rule: The instruction that turns every input into its own output.
flow diagram: A picture of a rule, drawn as a row of boxes with the input on the left.
inverse operation: The operation that undoes another. Subtracting undoes adding, and dividing undoes multiplying.

A rule turns one number into another. The number that goes in is the input, and the number that comes out is the output. A flow diagram draws that rule as a row of boxes.

Follow the boxes from left to right, in the order they are drawn. Not the order the words of the question happen to come in. If the multiply box is first, you multiply first.

Put a negative input in brackets before you do anything to it. $- 2$ squared is $4$ , because $(- 2)^{2} = 4$ . Written without brackets, $- 2^{2}$ reads as $- 4$ , which is a different number.

To find an input from an output, work backwards. Start at the last box and undo it with its inverse operation. Then undo the box before that one, and keep going to the left.

To find the rule from a table, compare each output with its own input. Do not compare it with the output above it. Then test the rule on every pair in the table, not just the first.

Rules to remember

Work forwards for an output, and backwards for an input

Examples

A flow diagram multiplies the input by

4

, then subtracts

3

. Find the output when the input is

- 2

Worked answer

Multiply first: $4 \times (- 2) = - 8$ .
Then subtract: $- 8 - 3 = - 11$ .
The output is $- 11$ .

Answer: $- 11$

The boxes were followed in the order drawn, so subtracting came last.

A rule multiplies the input by

4

, then subtracts

3

. The output is

17

. Find the input.

Worked answer

Start at the output and undo the last box first.
That box subtracted $3$ , so add $3$ : $17 + 3 = 20$ .
The first box multiplied by $4$ , so divide by $4$ : $20 \div 4 = 5$ .
The input is $5$ .
Check it forwards: $4 \times 5 - 3 = 17$ .

Answer: $5$

Each box was undone by its inverse operation, and in reverse order.

A table pairs the inputs

1; 2; 3

with the outputs

5; 9; 13

. Find the rule.

Worked answer

Each time the input rises by $1$ , the output rises by $4$ .
So the rule multiplies the input by $4$ .
Test that: $4 \times 1 = 4$ , but the first output is $5$ , so add $1$ .
The rule is $y = 4 x + 1$ .
Test the last pair too: $4 \times 3 + 1 = 13$ .

Answer: $y = 4 x + 1$

Every pair was tested, so a rule that fitted only the first was ruled out.

Traps

Applying a flow diagram’s operations in the order the words are written, not the order shown. Follow the diagram left to right exactly as drawn, even if the words come in a different order.
Squaring a negative input without brackets, so $- 2$ squared comes to $- 4$ . Put the input in brackets first: $(- 2)^{2} = 4$ .
Using the forward operations again instead of working backwards. To find an input, undo each operation with its opposite, in reverse order.
Keeping a rule that fits only the first pair in the table. Test the rule on every pair. For $y = 4 x + 1$ , an input of $3$ gives $13$ .

Learn and practise “Input and output values” in the app →

Equations, tables and graphs

One rule can be written five different ways, and all five have to agree with each other.

equivalent: Two rules are equivalent when every input gives them both the same output.
domain: The input values you are allowed to use, and no others.
Cartesian plane: The grid made by a flat $x$ -axis and an upright $y$ -axis crossing at zero.
ordered pair: One point written as two values, the $x$ -value first and the $y$ -value second.

The same rule can be given in words, as a flow diagram, as a table or as an equation. It can also be drawn as a graph. Five pictures, one rule. Half the work here is seeing that they match.

Never judge by how two rules look on the page. Put the same input into both and compare what comes out. Do that for several inputs before you call them equivalent.

A domain tells you which inputs to use. If the domain is $- 1$ , $0$ and $2$ , then you work out those three outputs and stop. Values outside the domain do not belong in your table.

A negative input needs brackets, especially inside a squared term or behind a minus sign. Write the brackets down before you do any arithmetic with it.

On the Cartesian plane every pair of values becomes a point. Plot the $x$ -value across first, then the $y$ -value up. An ordered pair written the wrong way round lands somewhere else.

Examples

Are "multiply by

2

, then add

6

" and "add

3

, then multiply by

2

" the same rule?

Worked answer

Try the input $4$ in the first rule: $2 \times 4 + 6 = 14$ .
Try the input $4$ in the second: $(4 + 3) \times 2 = 14$ .
Try the input $0$ : the first gives $6$ , and so does the second.
Try the input $- 1$ : both give $4$ .
Every input agrees, so the two rules are equivalent.

Answer: yes, they are equivalent

The rules look nothing alike, yet they do the same thing to every input.

Work out the outputs of

y = 3 x - 2

for the domain

- 1

0

and

2

Worked answer

Use only the three values the domain gives.
At $x = - 1$ : $3 (- 1) - 2 = - 5$ .
At $x = 0$ : $3 (0) - 2 = - 2$ .
At $x = 2$ : $3 (2) - 2 = 4$ .
The ordered pairs are $(- 1; - 5)$ , $(0; - 2)$ and $(2; 4)$ .

Answer: $- 5$ , $- 2$ and $4$

The domain fixed which inputs to use, and brackets kept the minus safe.

Which graph shows

y = 2 x - 4

: one through

(0; - 4)

that rises, or one through

(0; 4)

that falls?

Worked answer

Put $x = 0$ in to find where the graph cuts the $y$ -axis.
That gives $2 (0) - 4 = - 4$ .
So the graph has to pass through $(0; - 4)$ .
The number in front of $x$ is $2$ , which is positive, so the graph rises.
Only the first graph does both of those things.

Answer: the graph through $(0; - 4)$ that rises

Two features were checked, so the constant on its own did not decide it.

Traps

Deciding two rules must differ because their expressions look different. Put the same input into both. If every output matches, the rules are equivalent.
Working out outputs for values that were never in the domain. Use only the inputs the domain lists, and stop when they run out.
Losing the sign when a negative input is squared. Put a negative value in brackets before squaring it: $(- 3)^{2} = 9$ .
Reading a point as $(y; x)$ , with the two values the wrong way round. The $x$ -value always comes first. So $(3; 5)$ means $3$ across and $5$ up.

Learn and practise “Equations, tables and graphs” in the app →

Grade 9 Functions and relationships

What gets asked

You must be able to

Traps that cost marks

Worked example

Input and output values

Rules to remember

Examples

Traps

Equations, tables and graphs

Examples

Traps

About this material