Function Notation & Evaluation
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.
Practise this lesson
Three printable worksheets that build from foundations to mastery — or build your own from any module’s questions.
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?
$$f(a) \quad \text{replace every } x \text{ with } a$$
Key insight: when substituting algebraic expressions, always use brackets. This prevents sign errors.
Key facts
- how to evaluate $f(a)$ for numerical and algebraic inputs
- the meaning of the difference quotient
- how to interpret function notation in real-world contexts
Concepts
- that $f(x)$ describes a rule, not a multiplication
- why brackets are essential when substituting negatives or algebraic terms
- how functions model relationships like cost, distance, and temperature
Skills
- evaluate functions for numerical inputs, negative inputs, and zero
- evaluate functions for algebraic inputs such as $f(x+h)$ and $f(a)$
- set up and interpret functions in real-world contexts
- simplify and evaluate the difference quotient for linear and quadratic functions
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.
Numerical inputs
Suppose $f(x) = x^2 - 3x + 2$. To find $f(4)$:
- Replace $x$ with $4$: $f(4) = (4)^2 - 3(4) + 2$
- Simplify: $16 - 12 + 2 = 6$
Negative inputs
Negative inputs are a common source of errors. Always use brackets:
- $f(-2) = (-2)^2 - 3(-2) + 2$
- Simplify: $4 + 6 + 2 = 12$
Without brackets, $(-2)^2$ can easily become $-2^2 = -4$, which is incorrect. The bracket protects the sign.
Algebraic inputs
Functions can also be evaluated for algebraic expressions. If $f(x) = 2x + 1$, then:
- $f(a) = 2a + 1$
- $f(x+h) = 2(x+h) + 1 = 2x + 2h + 1$
Again, brackets are your best defence against mistakes. Every $x$ in the original rule must be replaced by the entire expression in parentheses.
Evaluate = replace every $x$ with the given value, then simplify; Always put brackets around substituted values: $f(-2) = (-2)^2$, not $-2^2$
Pause — copy the substitution rule (replace every $x$ with the input in brackets) and the negative-input bracket warning into your book.
Did you get this? True or false: to find $f(-3)$ for $f(x) = x^2$, you write $-3^2 = -9$.
Quick check: If $f(x) = 4x - 5$, what is $f(3)$?
We just saw that evaluating a function means substituting a value for $x$ and simplifying — even an algebraic expression like $x + h$ can be substituted. That raises a question: if we substitute $a + h$ and $a$ into the same function, what does the difference in outputs tell us? This card answers it → the difference quotient $\frac{f(a+h)-f(a)}{h}$ measures average rate of change over an interval.
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
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.
Context functions: the letter before brackets = output name; letter inside = input name (e.g. $C(d)$: cost depends on distance); $f(0)$ often represents a fixed fee, initial value, or starting point
Pause — copy the difference quotient formula $\dfrac{f(a+h)-f(a)}{h}$, its meaning (average rate of change), and the context-function naming convention into your book.
Fill the blanks: drag each token into the matching blank.
In $C(d) = 3 + 2d$, the distance $d$ is the ___. The $3$ is a ___ fee. The coefficient $2$ is the ___ of change. The cost $C$ is the ___.
Worked examples · reveal as you go
If $f(x) = 2x^2 - 3x + 1$, find $f(2)$, $f(-1)$, and $f(a)$.
A taxi charges a fare according to the rule $C(d) = 5 + 1.8d$, where $C$ is the cost in dollars and $d$ is the distance in kilometres. Find the cost of a 15 km trip and a 30 km trip. Interpret the meaning of $C(0)$.
For $f(x) = x^2 + 2x$, find $\dfrac{f(x+h) - f(x)}{h}$ in simplified form.
Common mistakes · the 4 traps that cost marks
Treating $f(x)$ as $f$ multiplied by $x$
This is the most common error in function notation. $f(x)$ means "the function $f$ evaluated at $x$" — it is not an algebraic product. Writing $f(3) = 3f$ or expanding $f(x+1) = f \cdot (x+1)$ shows a fundamental misunderstanding.
✓ Fix: Read $f(x)$ aloud as "$f$ of $x$." The parentheses contain the input, just like $g(2)$ or $h(a+b)$.
Forgetting brackets with negative or algebraic inputs
If $f(x) = x^2$ and you want $f(-2)$, writing $-2^2$ gives $-4$ because the exponent only applies to the $2$. The correct substitution is $(-2)^2 = 4$.
✓ Fix: Always write brackets around substituted values before simplifying: $f(-2) = (-2)^2$.
Only replacing the first occurrence of $x$
In $f(x) = x^2 - 3x + 1$, some students substitute correctly for the first $x$ but leave the second one unchanged, writing $f(a) = a^2 - 3x + 1$.
✓ Fix: Replace every instance of the variable in the rule. Count them before you simplify.
Errors when expanding $(x+h)^2$
A classic algebraic slip is writing $(x+h)^2 = x^2 + h^2$. The middle term $2xh$ is missing.
✓ Fix: Memorise the identity $(x+h)^2 = x^2 + 2xh + h^2$. It appears constantly in difference quotient problems.
Activity 1 — Evaluate and explain
Evaluate each function for the given input. Show your substitution step before simplifying.
If $f(x) = 4x - 5$, find $f(3)$ and $f(-2)$.
If $g(x) = x^2 - 3x + 2$, find $g(0)$ and $g(1)$.
If $h(x) = 2x^2 + x - 1$, find $h(a)$ and $h(x+h)$ in expanded form.
Odd one out: Which evaluation below contains an error?
Quick-fire practice · 5 reps +2 XP per reveal
Given $f(x) = 3x - 7$, find $f(4)$.
Given $g(x) = x^2 + 2x$, find $g(-3)$.
For $C(d) = 5 + 1.8d$, what does $C(0) = 5$ represent?
For $f(x) = 2x + 3$, show that the difference quotient $\dfrac{f(a+h)-f(a)}{h}$ equals $2$.
For $f(x) = x^2 - 4x + 3$, find $f(a+1)$ in simplified form.
Earlier you were asked: A taxi charges a $3 flag fall plus $2 for every kilometre. 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.
Pick your answer, then rate your confidence — that tells the system what to drill next.
Q8. 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)
Q9. 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)
Q10. 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)
📖 Comprehensive answers (click to reveal)
Multiple choice — drill bank
MC answers and feedback are shown inline as you complete each question. Use the retry button to attempt a fresh set.
Activity 1 — Evaluate and explain model answers
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. $h(a) = 2a^2 + a - 1$; $h(x+h) = 2(x+h)^2 + (x+h) - 1 = 2(x^2 + 2xh + h^2) + x + h - 1 = 2x^2 + 4xh + 2h^2 + x + h - 1$
Short answer model answers
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) $f(2) = (2)^2 - 4(2) + 3 = 4 - 8 + 3 = \mathbf{-1}$ ✓
(b) $f(-1) = (-1)^2 - 4(-1) + 3 = 1 + 4 + 3 = \mathbf{8}$ ✓
(c) $f(a+1) = (a+1)^2 - 4(a+1) + 3 = a^2 + 2a + 1 - 4a - 4 + 3 = \mathbf{a^2 - 2a}$ ✓
Award 1 mark each for (a), (b), and method in (c); 1 mark for correct simplified form in (c).
Q10 (4 marks):
(a) $f(a+h) = 2(a+h) + 3 = 2a + 2h + 3$
$\dfrac{f(a+h) - f(a)}{h} = \dfrac{(2a + 2h + 3) - (2a + 3)}{h} = \dfrac{2h}{h} = \mathbf{2}$ ✓
(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].
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