Page 44 - HSC Functions

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HSC Functions booklet — transformations, inverses, composition, and sketching with worked examples for Extension 1 and 2. Free online on Vu’s Maths Hub.

📚 Study Guide & Outline

1. HSC-Functions - an Introduction

Functions are the fundamental building blocks of almost all higher mathematics. If algebra is the language of maths, functions are the grammar. Historically, students underestimate this topic because it is introduced early in Year 11. However, a weak understanding of domains, ranges, and absolute value graphs will ruthlessly expose students in advanced calculus and curve sketching later in the HSC. This booklet bridges the gap between simple straight lines and the complex, dynamic graphs required in Extension 1 and 2. Mastering functions means you will avoid preventable domain restriction errors again.

2. Learning Outcomes & Syllabus Mapping

Working with Functions (Domain, Range, Notation).
Graphing Techniques (Transformations, Absolute Value).
Further Work with Functions (Inverse functions, Parametric forms).
Curve Sketching (Adding, multiplying, and dividing ordinates).
Solve absolute value inequalities analytically and graphically.
Determine the domains and ranges of composite and inverse functions.

3. Prerequisites

Mastery of basic algebraic expansion and factorization.
Familiarity with standard graphs (lines, parabolas, basic cubics, hyperbolas).
Understanding of basic inequalities.

4. Common HSC Mistakes

The most frequent mistake is ignoring implicit domain restrictions—specifically, forgetting that denominators cannot equal zero and you cannot take the even root of a negative number in the real plane. Another massive pitfall is solving absolute value inequalities like $|x - 2| > 3$ by simply dropping the modulus bars, completely missing the negative case and the resulting split domain.

5. Sample Worked Problem

Question: Find the domain of the function

f(x) = \frac{1}{\sqrt{x^2 - 4}}

Solution:

There are two restrictions here:

The denominator cannot be zero, so $x^2 - 4 \neq 0$ .
The term inside the square root must be strictly positive, so $x^2 - 4 > 0$ .

Solving the inequality: $x^2 > 4$ This splits into two regions on the number line: $x < -2$ or $x > 2$ . Therefore, the domain is all real $x$ such that $x < -2$ or $x > 2$ .

6. Exam Strategy & Weighting

Functions and graph transformations form the implicit foundation of about 40-50% of the HSC paper, though explicit "functions" questions account for a dedicated, smaller subset of the exam. You will frequently see multiple-choice questions asking you to identify the correct graph of an inverse function $y = f^{-1}(x)$ or an absolute value transformation $y = |f(x)|$ . The ability to quickly sketch a function to visualize an inequality will save you enormous amounts of time in Section II.

7. Key Definitions / Glossary Summary

Domain: The set of all possible input ( $x$ ) values for which the function is defined.
Range: The set of all possible output ( $y$ ) values produced by the function.
Absolute Value $|x|$ : The non-negative distance of $x$ from zero on the number line.
Asymptote: A line that a curve approaches, as it heads towards infinity, but never actually touches.

8. Frequently Asked Questions (FAQ)

Q: How do I find the range of a complicated function? A: The most reliable method is to sketch the graph using calculus (turning points, asymptotes). Alternatively, find the domain of its inverse function, as the domain of $f^{-1}(x)$ is the range of $f(x)$ .

Q: Are graphing calculators allowed in the HSC? A: No. You must be able to recognize and sketch standard functional transformations entirely by hand.

9. Where to next?

A rock-solid understanding of functions naturally leads into the study of rates of change. Your next step should be the HSC-DifferentialEquations booklet to see how functions behave dynamically, or HSC-Trigonometry to master periodic graphs.

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