How does a taxi meter know what to charge? It follows a simple rule: a fixed cost plus a rate for every kilometre travelled. In mathematics, we write this rule using function notation — and it opens the door to everything from economics to engineering.
Use the PDF for classwork, homework or revision. It includes key ideas, activities, questions, an extend task and success-criteria proof.
A taxi charges a $3 flag fall plus $2 for every kilometre travelled. How much would a 5 km trip cost? How much would a 10 km trip cost? Can you write a general rule using $C$ for cost and $d$ for distance?
Type your initial response below — you will revisit this at the end of the lesson.
Write your initial response in your book. You will revisit it at the end of the lesson.
Wrong: The domain of a function is always all real numbers.
Right: The domain depends on the function; rational functions exclude values that make the denominator zero.
📚 Core Content
To evaluate a function means to find the output for a given input. The process is always the same: replace every instance of the independent variable with the given value, then simplify using the correct order of operations.
Suppose $f(x) = x^2 - 3x + 2$. To find $f(4)$:
Negative inputs are a common source of errors. Always use brackets:
Without brackets, $(-2)^2$ can easily become $-2^2 = -4$, which is incorrect. The bracket protects the sign.
Functions can also be evaluated for algebraic expressions. If $f(x) = 2x + 1$, then:
Again, brackets are your best defence against mistakes. Every $x$ in the original rule must be replaced by the entire expression in parentheses.
Function notation becomes powerful when we use it to model real situations. A taxi fare might be written as $C(d) = 3 + 2d$, where $C$ is the cost in dollars and $d$ is the distance in kilometres. The notation tells us instantly what the variables represent and how they are related.
The difference quotient measures the average rate of change of a function over an interval:
$$\frac{f(a+h) - f(a)}{h}$$
This expression appears throughout calculus. For now, think of it as the average slope of the function between the points $x = a$ and $x = a+h$. If the difference quotient is constant for all values of $a$ and $h$, the function is a straight line.
🧮 Worked Examples
🧪 Activities
1 If $f(x) = 4x - 5$, find $f(3)$ and $f(-2)$.
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2 If $g(x) = x^2 - 3x + 2$, find $g(0)$ and $g(1)$.
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3 If $h(x) = 2x^2 + x - 1$, find $h(a)$ and $h(x+h)$ in expanded form.
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1 A gym charges a $20 joining fee plus $15 per week. Write a function $C(w)$ for the total cost after $w$ weeks, and find the cost for 12 weeks.
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2 The temperature of a cooling object is given by $T(t) = 100 - 5t$, where $T$ is in degrees Celsius and $t$ is in minutes. Find $T(10)$ and explain what it means in context.
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3 A phone plan costs $P(m) = 30 + 0.5m$ dollars for $m$ minutes of international calls. What does $P(0)$ represent? What is the rate per minute?
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Earlier you were asked: A taxi charges a $3 flag fall plus $2 for every kilometre travelled. How much would a 5 km trip cost? How much would a 10 km trip cost? Can you write a general rule?
A 5 km trip costs $3 + 2(5) = \\$13, and a 10 km trip costs $3 + 2(10) = \\$23. The general rule is $C(d) = 3 + 2d$, where $C$ is the cost in dollars and $d$ is the distance in kilometres. This is a function because each distance input produces exactly one cost output. The $3$ is a constant (the flag fall) and the $2$ is the rate of change per kilometre — ideas that will follow you through calculus and beyond.
Now revisit your initial response. What did you get right? What has changed in your thinking?
Look back at your initial response in your book. Annotate it with what you now understand differently.
Look back at what you wrote in the Think First section. What has changed? What did you get right? What surprised you?
5 random questions from a replayable lesson bank — feedback shown immediately
✍️ Short Answer
8. Explain the difference between $f(x)$ and $f \times x$. Use a specific example with $f(x) = 2x + 3$ to illustrate your answer. 2 MARKS
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9. Consider the function $f(x) = x^2 - 4x + 3$. (a) Find $f(2)$. (b) Find $f(-1)$. (c) Find $f(a+1)$ in simplified form. Show all working. 4 MARKS
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10. The difference quotient $\dfrac{f(a+h) - f(a)}{h}$ measures the average rate of change of a function. (a) For $f(x) = 2x + 3$, show that the difference quotient equals $2$. (b) Explain what this result tells you about the graph of $f(x) = 2x + 3$. 4 MARKS
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Answer in your workbook.
1. $f(3) = 4(3) - 5 = 12 - 5 = 7$; $f(-2) = 4(-2) - 5 = -8 - 5 = -13$
2. $g(0) = (0)^2 - 3(0) + 2 = 2$; $g(1) = (1)^2 - 3(1) + 2 = 1 - 3 + 2 = 0$
3. $h(a) = 2a^2 + a - 1$; $h(x+h) = 2(x+h)^2 + (x+h) - 1 = 2(x^2 + 2xh + h^2) + x + h - 1 = 2x^2 + 4xh + 2h^2 + x + h - 1$
1. $C(w) = 20 + 15w$; $C(12) = 20 + 15(12) = 20 + 180 = \\$200$
2. $T(10) = 100 - 5(10) = 50^\circ$C. After 10 minutes, the object has cooled to $50^\circ$C.
3. $P(0) = 30 + 0.5(0) = 30$. This represents the fixed monthly cost of $30 before any international calls are made. The rate per minute is $0.50.
1. A — $f(4) = 3(4) - 7 = 5$
2. B — The cost is $5 plus $1.50 for every kilometre travelled.
3. B — $f(-3) = (-3)^2 + 2(-3) = 9 - 6 = 3$
4. B — $f(x+h) = 2(x+h) + 1 = 2x + 2h + 1$
5. B — The expression is the difference quotient.
Q8 (2 marks): $f(x)$ is function notation meaning "the output of function $f$ when the input is $x$" [1]. In contrast, $f \times x$ means the variable $f$ multiplied by $x$. For example, with $f(x) = 2x + 3$, we have $f(4) = 2(4) + 3 = 11$, which is completely different from $f \times 4$ [1].
Q9 (4 marks):
(a) $f(2) = (2)^2 - 4(2) + 3 = 4 - 8 + 3 = -1$ (b) $f(-1) = (-1)^2 - 4(-1) + 3 = 1 + 4 + 3 = 8$ (c) $f(a+1) = (a+1)^2 - 4(a+1) + 3 = a^2 + 2a + 1 - 4a - 4 + 3 = a^2 - 2a$Award 1 mark each for (a), (b), and method in (c); 1 mark for correct simplified form in (c).
Q10 (4 marks):
(a) $f(a+h) = 2(a+h) + 3 = 2a + 2h + 3$ $\frac{f(a+h) - f(a)}{h} = \frac{(2a + 2h + 3) - (2a + 3)}{h} = \frac{2h}{h} = 2$(b) The difference quotient equals 2 for all values of $a$ and $h$ [1]. This tells us that the average rate of change is constant [1], which means the graph of $f(x) = 2x + 3$ is a straight line with a slope of 2 [1].
Sprint through questions on function notation and evaluation. Pool: lessons 1–2.
Tick when you've finished all activities and checked your answers.