Welcome to Functions & Exponentials — the heart of algebra and one of the most tested topics on the CSCA exam. A function is like a machine: you feed it an input, it applies a rule, and it gives you an output. Understanding functions means understanding how things change.
In this module, you will learn to describe functions through their domain and range, handle piecewise definitions, determine symmetry and injectivity, find inverses and compositions, and then dive deep into exponential and logarithmic functions — the tools that model everything from population growth to radioactive decay. Finally, you will master transformations that shift, stretch, and reflect graphs.
Let's get started!
The CSCA exam typically includes 5-7 questions on functions and exponentials, worth 20-30 marks total. Expect: domain/range problems, inverse function calculations, composition evaluations, exponential/logarithmic equation solving, growth/decay word problems, and transformation identification. Master the log rules and exponential laws — they unlock most of these questions.
The domain of a function f is the set of all valid inputs (x-values). The range is the set of all possible outputs (y-values or f(x)-values).
Finding the Domain
Ask: "What values of x would cause a problem?" Common restrictions:
Finding the Range
Methods include:
Notation: Domain and range are sets, so express them using set or interval notation.
Think of the domain as the "menu" of a restaurant (what you can order) and the range as the "dishes that actually come out of the kitchen" (what you actually get).
💡Domain = all valid inputs (avoid ÷0, √negative, log of non-positive). Range = all possible outputs. Express both in interval notation.
📋 Key Formulas
√(expr) needs expr ≥ 0 | 1/expr needs expr ≠ 0 | log(expr) needs expr > 0
📝 Worked Example 1
Example 1: Find the domain of f(x) = √(6 - 2x).
Need 6 - 2x ≥ 0 → -2x ≥ -6 → x ≤ 3. Domain: (-∞, 3].
📝 Worked Example 2
Example 2: Find the domain and range of f(x) = 1/(x² + 1).
Domain: x² + 1 ≥ 1 > 0 for all x, so no restrictions. Domain = ℝ.
Range: Max value is 1/(0+1) = 1 (at x=0). As x → ±∞, f → 0⁺. Range = (0, 1].
📝 Worked Example 3
Example 3: Find the domain of g(x) = ln(x² - 4).
Need x² - 4 > 0 → x² > 4 → |x| > 2 → x < -2 or x > 2.
Domain: (-∞, -2) ∪ (2, ∞).
🧠For domain, systematically check for division by zero, square roots, and logarithms.
🧠For range, try the algebraic method: set y = f(x), solve for x, and see which y-values produce real x.
⚠️Forgetting to exclude log argument = 0: ln(0) is undefined, so you need strict inequality x > 0, not x ≥ 0.
⚠️Saying range of x² is ℝ: x² ≥ 0 always, so range is [0, ∞), not all reals.
🎯 Try This Yourself
Find the domain of f(x) = √(x+3) / (x-1).
In this module you mastered functions from the ground up: domain and range, piecewise definitions, symmetry, injectivity, inverses, and compositions. You then explored the world of exponentials and logarithms — from basic properties to growth/decay models and equation-solving techniques. Finally, you learned to transform any function graph through shifts, stretches, and reflections. These skills are foundational for calculus and appear throughout the CSCA exam.
Open and read all sections to complete this module