Course Summary
This course is built around “the function”, the core concept of advanced mathematics: it first establishes the basic concepts of a function (its domain, its range, and the ways it can be represented), then studies the various properties of functions, then moves from the special function forms such as composite and inverse functions to the complete system of elementary functions, and finally closes with the common constants such as , , and the golden ratio. Looking back over the whole course, the main thread is: from “what a function is” to “what properties a function has”, and on to “how the elementary functions are built up”.
| Topic | Core content | Key methods |
|---|---|---|
| Basic Concepts of Functions | The definition of a function, its domain, its range, and its representations | The formula, the graph, and the table; when finding the domain, exclude the points where the expression is meaningless |
| Properties of Functions | Monotonicity, parity, periodicity, boundedness, extrema and extreme values, convexity and inflection points | The definition, graph observation, and derivative tests (connecting to the later differential calculus course) |
| Special Function Forms | Composite functions, inverse functions, piecewise functions, implicit functions, and parametric functions | Splitting composites, solving inverse functions, discussing the piecewise points, and handling the implicit and parametric forms |
| Elementary Functions | The basic elementary functions: power, exponential, logarithmic, trigonometric, and inverse trigonometric functions | Combine the formula with the graph, and master the domain, the range, and the characteristic behaviour of each |
| Common Constants and Functions | Irrational functions, rational functions, , , and the golden ratio | Understand where the constants come from, and recognise the graphs and the properties of the typical functions |
The properties of functions and the key points of the special function forms are the two threads that run through this course:
| Property/Concept | Core result | How to test it |
|---|---|---|
| Monotonicity | On an interval the function increases (or decreases) as the independent variable increases | Compare by taking the difference, using the definition; or use the sign of the derivative (connecting to differential calculus) |
| Parity | An odd function satisfies , an even function satisfies | First check whether the domain is symmetric about the origin, then substitute and compare |
| Periodicity | There is a with , and the smallest positive period is the fundamental period | The period formulas of the trigonometric functions, for example and have the period |
| Boundedness | There is an with holding throughout the domain | Estimate from the ranges of elementary functions, for example |
| Composite Functions | Composing with gives | The range of the inner function must fall inside the domain of the outer one |
| Inverse Functions | If is one-to-one, then | Solve for and then swap and ; note that the domain and the range are exchanged |
Key takeaway: When judging the properties of a function, always start from the domain; parity, periodicity, and boundedness often mean “look at the graph first, then verify it strictly”; an elementary function is obtained from the basic elementary functions by a finite number of arithmetic operations and compositions, and is the cornerstone for studying all other functions.
When you meet a question about functions, think along the following path:
- Clarify what the question is about: are you after the domain, the range, a judgement about some property, an inverse function, or a composite function? Settle the question first and then start.
- Finding the domain comes first: the denominator must be non-zero, the radicand of an even root must be non-negative, the argument of a logarithm must be positive, and the argument of an inverse trigonometric function must lie in its own range.
- Property questions go back to the definition: monotonicity, parity, and periodicity are all verified with the definition; do not jump to conclusions from the graph, and check the symmetry of the domain before verifying parity.
- Mind the order in composites and inverses: in a composite function compute the inner function first and the outer one second; after finding an inverse function, swap the variables and exchange the domain with the range.
- Be familiar with the graphs of the basic elementary functions: the graphs and the key points of the power, exponential, logarithmic, trigonometric, and inverse trigonometric functions are the basis for analysing the properties of complicated functions, and they help you rule out wrong answers quickly.
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Exploring Functions in Advanced Mathematics
Current courseFunctions are a core idea of advanced mathematics. This course walks through foundational concepts, key properties, and classic constants so you can read, reason, and compute with confidence.
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