Function notation and evaluating rules | Year 11

01 / LEARN

Understand the idea.

A rule with named inputs

A function assigns exactly one output to each allowed input. In $f (3)$ , the brackets identify the input $3$ ; the notation means the output of the rule named $f$ . It is not a multiplication.

Evaluate a rule by replacing every occurrence of its input variable. Write negative or compound inputs in brackets first: squaring $(- 2)$ gives $4$ , whereas $- 2^{2}$ means the negative of $2^{2}$ .

f (x) = 2 x^{2} - 3 x + 1

Values of f(x)=2x²-3x+1
x	f(x)
-2	15
-1	6
0	1
1	0
2	3

Inputs can be expressions

An algebraic expression can also be an input. To evaluate $f (a + 1)$ , replace every $x$ by the whole expression $a + 1$ , keeping brackets until the substitution is complete. Then expand and collect terms if required.

Evaluating $f (3)$ starts with an input and finds an output. Solving $f (x) = 7$ starts with a target output and finds the possible inputs. Either operation must respect the stated domain; a short table alone does not determine a unique rule.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A negative input

For $f (x) = 2 x^{2} - 3 x + 1$ , find $f (- 2)$ .

Worked solution

1
Substitute the complete input.
$f (- 2) = 2 {(- 2)}^{2} - 3 (- 2) + 1$
Brackets ensure the negative sign is included in the number being squared.
2
Evaluate the power and signed products.
$f (- 2) = 2 (4) + 6 + 1$
The square is positive, and multiplying two negative numbers gives a positive number.
3
Add the terms.
$f (- 2) = 8 + 6 + 1 = 15$
The output is a single number after every occurrence of the input has been replaced.

Answer

$15$

EXAMPLE 02

An expression as input

If $g (x) = 3 x - 4$ , simplify $g (t + 2)$ and solve $g (x) = 11$ .

Worked solution

1
Replace the input by the whole expression.
$g (t + 2) = 3 (t + 2) - 4$
The rule multiplies its entire input by three before subtracting four.
2
Expand and collect constants.
$g (t + 2) = 3 t + 6 - 4 = 3 t + 2$
The distributive law multiplies both terms inside the brackets.
3
Set the rule equal to the target output.
$\begin{aligned} 3 x - 4 & = 11 \\ 3 x & = 15 \\ x & = 5 \end{aligned}$
Adding four and dividing by three reverse the operations in the rule.
4
Check the recovered input.
$g (5) = 3 (5) - 4 = 11$
Substitution confirms that the input five produces the required output.

Answer

$g (t + 2) = 3 t + 2$ ; $x = 5$

03 / YOUR TURN

Try it yourself.

Check your understanding

If $h (x) = x^{2} + 2 x$ , what is $h (- 3)$ ?

Show answer & reasoning

$3$

Replace both occurrences of the input by $- 3$ : $h (- 3) = {(- 3)}^{2} + 2 (- 3) = 9 - 6 = 3$ . The square and the linear term have different signs.

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

For $f (x) = 4 x - 7$ , find $f (3)$ .
2 marks
For $f (x) = x^{2} - 5 x$ , find $f (- 2)$ .
2 marks
If $p (x) = 2 x + 5$ , simplify $p (a - 3)$ .
2 marks
Find the input for which $g (x) = 7 x + 2$ has output 30.
2 marks
Let $r (x) = \frac{12}{x + 1}$ . Find $r (2)$ and state an input that cannot be used.
2 marks
A delivery fee is $C (d) = 5 + 1.8 d$ dollars for $d$ kilometres. Find and interpret $C (10)$ .
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

For $f (x) = 4 x - 7$ , find $f (3)$ .
2 marks
$f (3) = 4 (3) - 7 = 12 - 7 = 5$ .
$5$
For $f (x) = x^{2} - 5 x$ , find $f (- 2)$ .
2 marks
$f (- 2) = {(- 2)}^{2} - 5 (- 2) = 4 + 10 = 14$ .
$14$
If $p (x) = 2 x + 5$ , simplify $p (a - 3)$ .
2 marks
$p (a - 3) = 2 (a - 3) + 5 = 2 a - 1$ .
$2 a - 1$
Find the input for which $g (x) = 7 x + 2$ has output 30.
2 marks
Solve $7 x + 2 = 30$ .
Then $7 x = 28$ , so $x = 4$ .
$x = 4$
Let $r (x) = \frac{12}{x + 1}$ . Find $r (2)$ and state an input that cannot be used.
2 marks
$r (2) = \frac{12}{3} = 4$ .
At $x = - 1$ the denominator is zero.
$r (2) = 4$ ; $x = - 1$ is excluded.
A delivery fee is $C (d) = 5 + 1.8 d$ dollars for $d$ kilometres. Find and interpret $C (10)$ .
2 marks
$C (10) = 5 + 1.8 (10) = 23$ .
The output is the total fee in dollars, not the distance.
$23$ for a 10 km delivery.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics, Topic 1: Functions and graphs. Prerequisite algebra and coordinate geometry are embedded as supporting lessons.

Official curriculum source ↗

In this lesson

Understand the idea.

A rule with named inputs

Inputs can be expressions

See the method in action.

A negative input

Substitute the complete input.

Evaluate the power and signed products.

Add the terms.

An expression as input

Replace the input by the whole expression.

Expand and collect constants.

Set the rule equal to the target output.

Check the recovered input.

Try it yourself.

Check your understanding

Keep practising