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

Finding input and output values, and the rule that connects them.

Gr 8 Terms 1 and 4 — Functions and relationshipsTerm 1 and Term 4

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What gets asked

Find an output value using a flow diagram, table or formula.
Find an input value when the output and the rule are given.
Write the rule that connects the input and output values.
Decide whether two rules describe the same relationship.

You must be able to

Apply every operation in a two-step flow diagram, in order.
Work backwards using inverse operations to find an input value.
Test a rule against every pair of values in a table before trusting it.
Substitute given values into a formula to calculate an unknown one.

Traps that cost marks

Applying only the first operation of a two-step flow diagram and stopping there. A two-step flow diagram needs both operations, done in order, every time.
Undoing the two operations of a rule in the same order they were applied, instead of reverse order. To work backwards, undo the last operation first, then the first one.
Finding a rule that fits only the first pair of input and output values. Check your rule against every pair of values in the table, not just one.
Multiplying by a known value instead of dividing when finding an unknown input from a formula. Work out which operation the formula uses, then apply the opposite to undo it.

Worked example

A flow diagram: input

\to \times 3 \to + 4 \to

output. If the output is 22, find the input.

Undo $+ 4$ first: $22 - 4 = 18$ .
Undo $\times 3$ next: $18 \div 3 = 6$ .
The input is 6.

Flow diagrams and rules

Following a rule forwards to get an answer, and backwards to find where the answer came from.

input: The number you put into a rule.
output: The number that comes out once the rule has been used.
rule: The instruction that changes every input into its output.
flow diagram: A row of boxes that draws a rule, with one step in each box.
inverse operation: The operation that undoes another one. Adding undoes taking away.

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

Work through the boxes from left to right, and use every box. If the rule multiplies by $3$ and then adds $4$ , an input of $5$ gives $15$ , and then $19$ . Stopping at $15$ is only half an answer.

Put a negative input in brackets first. With the same rule and an input of $- 2$ : $3 \times (- 2) = - 6$ , then $- 6 + 4 = - 2$ . The brackets are what keep the minus sign from getting lost.

To find an input from an output, go backwards. Start at the last box and use the inverse operation. Then undo the box before it. The wrong order gives the wrong input.

To find a rule from a table, compare each output with its own input, not with the output next to it. Two rules that look different can still be the same rule. Test both on the same inputs.

Rules to remember

Forwards for an output. Backwards, in reverse order, for an input.

Examples

A flow diagram multiplies the input by

3

, then adds

4

. Find the output when the input is

- 2

Worked answer

Take the boxes in order, so multiply first.
$3 \times (- 2) = - 6$ .
Now add: $- 6 + 4 = - 2$ .

Answer: $- 2$

Both boxes were used, and the brackets kept the minus sign safe.

The same rule gives an output of

22

. Find the input.

Worked answer

Undo the last box first, so take away $4$ : $22 - 4 = 18$ .
Undo the multiply next: $18 \div 3 = 6$ .
Check it forwards: $3 \times 6 = 18$ , then $18 + 4 = 22$ .
So the input is $6$ .

Answer: $6$

The last box was undone first, and that is what reverse order means.

A table pairs the inputs

1; 2; 3

with the outputs

6; 11; 16

. Find the rule.

Worked answer

Compare each output with its own input, not with the output beside it.
The output climbs by $5$ each time the input climbs by $1$ , so start with $5 x$ .
At $x = 1$ that gives $5$ , but the output is $6$ , so add $1$ .
Test the other pairs: $5 (2) + 1 = 11$ and $5 (3) + 1 = 16$ .

Answer: $y = 5 x + 1$

All three pairs were tested, so the rule is not just a lucky fit.

Traps

Applying only the first operation of a two-step flow diagram and stopping there. A two-step flow diagram needs both operations, done in order, every time.
Undoing the two operations of a rule in the same order they were applied, instead of reverse order. To work backwards, undo the last operation first, then the first one.
Finding a rule that fits only the first pair of input and output values. Check your rule against every pair of values in the table, not just one.
Marking a rule wrong because it is written differently from your own. Try both rules on the same inputs. If every output agrees, both rules are correct.

Learn and practise “Flow diagrams and rules” in the app →

Using formulae

Turning a sentence into a formula, using it both ways, and testing whether two rules agree.

formula: A rule written with letters and an equal sign, such as $c = 12 n$ .
equation: Two amounts joined by an equal sign, saying they are worth the same.
substitute: To write a number in place of a letter, then work the answer out.
equivalent: Two rules are equivalent when they give the same output for every input.

A formula writes a rule in letters. "Three times a number, then add one" becomes $y = 3 x + 1$ . Follow the words in the order they come. Multiply first here, because that is what the words do first.

A formula needs an equal sign. On its own, $3 x + 1$ is only an expression. It never says what the answer is called. Writing $y = 3 x + 1$ names it, so now you have an equation.

To use a formula, check which letter each number belongs to. Then substitute and work it out. Give the answer with its unit. When two lengths are multiplied, the unit is squared.

To find an input, undo the formula with inverse operations. If a taxi trip costs R25 and the formula is $c = 25 n$ , do not multiply R300 by $25$ . Divide instead: $300 \div 25 = 12$ .

Two rules can look different and still be equivalent. Test $y = 2 (x + 3)$ against $y = 2 x + 6$ . At $x = 1$ both give $8$ . At $x = 4$ both give $14$ . One test is never enough.

Rules to remember

Rules are equivalent when they agree for every input, not just for one.

Examples

Airtime vouchers cost R12 each, and delivery costs R15 on top. Write a formula for the cost

c

n

vouchers.

Worked answer

Each voucher multiplies, so the vouchers cost $12 n$ .
Delivery is added once, so add $15$ .
Name the total and join it with an equal sign.

Answer: $c = 12 n + 15$

Without the equal sign this would be an expression, not a formula.

A garden plot is a rectangle with area

A = l \times b

. Find

A

when

l = 9

m and

b = 4

Worked answer

Match each number to its own letter, so $l$ takes $9$ and $b$ takes $4$ .
Substitute into the formula: $A = 9 \times 4$ .
Work it out: $9 \times 4 = 36$ .
Two lengths were multiplied, so the unit is squared.

Answer: $36$ m $^{2}$

The number came from the formula and the unit came from the measuring.

Taxi trips cost R25 each, so

c = 25 n

. Thabo spent R300. How many trips did he take?

Worked answer

The R300 is a cost, so it takes the place of $c$ , not of $n$ .
The formula multiplies by $25$ , so undo it by dividing.
$300 \div 25 = 12$ .
Check it forwards: $25 \times 12 = 300$ .

Answer: $12$ trips

Dividing is the inverse operation of multiplying, so it found the input.

Traps

Writing "three times a number, then add one" as $3 (x + 1)$ . The words multiply first and add second, so the formula is $y = 3 x + 1$ .
Giving the area of a $9$ m by $4$ m plot as $36$ m. Two lengths were multiplied, so the area is $36$ m $^{2}$ .
Multiplying by a known value instead of dividing when finding an unknown input from a formula. Work out which operation the formula uses, then apply the opposite to undo it.
Trying one pair of values and calling two rules the same. Test two or three inputs. Two rules can agree once by accident.

Learn and practise “Using formulae” in the app →

Grade 8 Functions and relationships

What gets asked

You must be able to

Traps that cost marks

Worked example

Flow diagrams and rules

Rules to remember

Examples

Traps

Using formulae

Rules to remember

Examples

Traps

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