Introduction to Extension 1 Functions
You already know what a function is — an input, a rule, an output. But Extension 1 asks a deeper question: what happens when we flip that rule around? What does it mean for two expressions to be constrained by inequality rather than equality? This module unlocks four powerful function tools — inequalities, inverses, graphical transformations, and parametric equations — that will reappear in every calculus and proof topic to come.
What is the difference between a relation and a function? Write down the definition of domain and range, and explain how you would use the vertical line test. Don't look anything up — write what you already know.
Everything in Extension 1 Functions flows from two core ideas. First: test whether a relation is a function using the vertical line test. Second: test whether a function is one-to-one using the horizontal line test — because only one-to-one functions have inverses.
The vertical line test asks: does any vertical line cross the graph more than once? If yes, it's not a function. The horizontal line test asks: does any horizontal line cross the graph more than once? If yes, the function is many-to-one and has no inverse function over that domain.
Key facts
- The four key areas of this module: inequalities, inverse functions, graphical relationships, parametric equations
- Domain and range of common functions: $x^2$, $\sqrt{x}$, $\frac{1}{x}$, $e^x$, $\ln x$
- A function has an inverse only when it is one-to-one
Concepts
- How Extension 1 extends Advanced function concepts into inequalities and inverses
- Why the horizontal line test determines whether an inverse function exists
- How domain restrictions allow non-one-to-one functions to have inverses
Skills
- State the domain and range of standard functions
- Apply vertical and horizontal line tests to graphs
- Identify appropriate domain restrictions for inverse functions
In Mathematics Advanced, you studied functions and their transformations: translations, reflections and dilations. In Extension 1, the module Further Work with Functions goes much deeper into four interconnected areas:
- Inequalities — solving linear, quadratic, rational and absolute value inequalities algebraically and graphically.
- Inverse functions — finding inverses algebraically, understanding when domains must be restricted, and sketching inverse graphs.
- Graphical relationships — sketching $y = |f(x)|$, $y = f(|x|)$, $y = \dfrac{1}{f(x)}$, $y = \sqrt{f(x)}$ and $y = [f(x)]^2$ from the graph of $y = f(x)$.
- Parametric equations — describing curves using a parameter, eliminating the parameter, and sketching parametric curves.
All four areas are essential for the HSC Extension 1 examination and form the foundation for calculus and proof in later modules.
Four key areas: (1) Inequalities, (2) Inverse functions, (3) Graphical relationships, (4) Parametric equations; A function has an inverse function only if it is one-to-one (passes the horizontal line test)
Pause — copy the four key areas of Module 1 into your book — inequalities, inverse functions, graphical transformations, parametric equations — and the horizontal line test condition for inverse functions.
Quick check: Which of the following is NOT one of the four key areas in the Extension 1 Functions module?
We just saw that Module 1 covers four areas: inequalities, inverse functions, graphical transformations, and parametric equations. That raises a question: what prerequisite function knowledge — domains, ranges, function notation — do you need before tackling any of these four areas? This card answers it → by reviewing the essential toolkit: domain, range, the horizontal line test, and the key functions $sqrt{x}$ and $1/x$.
Before diving into new material, recall the essential function concepts from Advanced:
- A function is a relation where every input ($x$-value) has exactly one output ($y$-value).
- The domain is the set of all possible inputs.
- The range is the set of all possible outputs.
- A function is one-to-one if every output comes from exactly one input (passes the horizontal line test).
Key domains and ranges to memorise:
| Function | Domain | Range |
|---|---|---|
| $f(x) = x^2$ | All real $x$ | $y \ge 0$ |
| $f(x) = \sqrt{x}$ | $x \ge 0$ | $y \ge 0$ |
| $f(x) = \dfrac{1}{x}$ | $x \ne 0$ | $y \ne 0$ |
| $f(x) = e^x$ | All real $x$ | $y > 0$ |
| $f(x) = \ln x$ | $x > 0$ | All real $y$ |
Memory rule for domains: square roots need $\ge 0$ under the root; denominators cannot be zero; logarithms need a positive argument.
Domain: set of allowed inputs. Range: set of possible outputs.; x: domain x 0, range y 0. 1{x}: domain x 0, range y 0.
Pause — copy the domain and range of the key functions into your book: $\sqrt{x}$ has domain $x \ge 0$, range $y \ge 0$; $1/x$ has domain $x \ne 0$, range $y \ne 0$.
Did you get this? True or false: the function $f(x) = x^3$ is one-to-one over all real $x$.
Worked examples · 3 in a row, reveal as you go
State the domain and range of $f(x) = \sqrt{x - 3}$.
Show that $f(x) = x^3$ is one-to-one, but $g(x) = x^2$ is not.
The 15 lessons in this module are grouped into four areas. Describe each area and the lesson range it covers.
Fill the gap: The domain of $f(x) = \sqrt{x - 5}$ is $x \ge $ .
Misconceptions to fix · the 3 traps that cost marks
Did you get this? True or false: $f(x) = x^2$ over the domain $x \ge 0$ has an inverse function.
Odd one out: Which of the following is the odd one out? Select the function with a different property from the rest.
Activities · practice with the ideas
State the domain and range of $f(x) = \dfrac{1}{x - 2}$.
Is $f(x) = \ln x$ one-to-one? Explain your reasoning and state whether it has an inverse.
Explain why $f(x) = x^2$ does not have an inverse function over its natural domain. State a restricted domain for which an inverse does exist.
List the four key topic areas studied in this module. Which area do you think will be most challenging? Give a reason.
Determine whether each of these is a function and, if so, whether it is one-to-one: (a) $y = x^2 - 4$, (b) $x = y^2$, (c) $y = e^x$.
Earlier you were asked: if $f(x) = x^2$, what is $f^{-1}(9)$?
Over the natural domain (all real $x$), $f(x) = x^2$ is not one-to-one — both $x = 3$ and $x = -3$ give $f(x) = 9$. So $f^{-1}$ does not exist without a domain restriction. If we restrict to $x \ge 0$, then $f^{-1}(9) = 3$. If we restrict to $x \le 0$, then $f^{-1}(9) = -3$. The domain restriction determines which inverse you get.
Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
Q1. State the domain and range of $f(x) = \dfrac{1}{x - 2}$. (2 marks)
Q2. Explain why $f(x) = x^2$ does not have an inverse function over its natural domain, and state a restricted domain for which an inverse does exist. (2 marks)
Q3. List the four key topic areas studied in the Extension 1 Functions module, and briefly describe what is covered in each. (4 marks)
Comprehensive answers (click to reveal)
Activity answers: 1. Domain $x \ne 2$, range $y \ne 0$. · 2. $\ln x$ is one-to-one (always increasing, horizontal line test passes); inverse is $e^x$. · 3. $x^2$ fails horizontal line test ($g(2) = g(-2) = 4$); restrict to $x \ge 0$ or $x \le 0$. · 4. (1) Inequalities, (2) Inverse functions, (3) Graphical relationships, (4) Parametric equations. · 5. (a) $y = x^2 - 4$: function, not one-to-one. (b) $x = y^2$: not a function (vertical line test fails). (c) $y = e^x$: function, one-to-one.
Q1 (2 marks): Domain: $x \ne 2$ [or $(-\infty, 2) \cup (2, \infty)$] [1 mark]. Range: $y \ne 0$ [or $(-\infty, 0) \cup (0, \infty)$] [1 mark].
Q2 (2 marks): $f(x) = x^2$ fails the horizontal line test — for example $f(2) = f(-2) = 4$, so two inputs give the same output [1 mark]. A restricted domain such as $x \ge 0$ makes $f$ one-to-one and its inverse $f^{-1}(x) = \sqrt{x}$ exists [1 mark].
Q3 (4 marks): (1) Inequalities — solving $<$, $>$, $\le$, $\ge$ statements algebraically and graphically [0.5+0.5]. (2) Inverse functions — finding $f^{-1}$, restricting domains, using $f^{-1}(f(x)) = x$ [0.5+0.5]. (3) Graphical relationships — sketching $|f(x)|$, $f(|x|)$, $1/f(x)$, $\sqrt{f(x)}$, $[f(x)]^2$ [0.5+0.5]. (4) Parametric equations — expressing $x$ and $y$ in terms of parameter $t$, eliminating the parameter [0.5+0.5].
Five timed questions. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
⚔ Enter the arenaClimb platforms by answering Extension 1 functions questions. Lighter alternative to the boss.
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