Precalculus
Logarithms Explained: Definition, Laws and Solving Equations
A logarithm answers one question: what power do I raise the base to in order to get this number? Every rule and technique in this guide follows from that single idea.
Precalculus
Precalculus is about functions: what they accept, what they return, how they combine and how to undo them. These guides build the toolkit that calculus assumes you already have.
Calculus courses move quickly, and most of the difficulty students report there is really precalculus difficulty: an inverse function found incorrectly, a domain restriction forgotten, or a logarithm rule applied to a sum. The articles in this section slow those topics down.
You will find clear definitions, the rules written as formulas, worked examples with every step, and checks you can run on your own answers, such as composing a function with its inverse to confirm you get back . Related calculators let you test the same examples and try your own.
Precalculus also introduces the habit of reading a function three ways: as a formula, as a table of values and as a graph. Several articles show all three views of the same function, because switching between them is often the quickest way to spot a wrong answer or an impossible domain.
A good order is domain and range first, then composition and inverses, then logarithms and polynomial functions.
All articles in this category (5)
Precalculus
How to Find the Domain and Range of a Function
The domain is every input a function accepts; the range is every output it can produce. Finding the domain means hunting for forbidden inputs, and finding the range means asking which outputs are actually reachable.
Precalculus
Function Composition: How to Find and Use f(g(x))
Composing functions means feeding the output of one function into another. It is how complicated formulas are built from simple pieces, and recognising those pieces is essential for inverses and for the chain rule in calculus.
Precalculus
Inverse Functions: How to Find, Graph and Verify an Inverse
An inverse function undoes what the original function does. If $f$ sends 2 to 7, its inverse sends 7 back to 2. This guide explains when an inverse exists, how to find its formula, and how to check it.
Precalculus
Polynomial Functions: Degree, End Behaviour, Zeros and Graphs
Polynomials are the simplest functions beyond lines and parabolas, and their graphs follow a few dependable rules. Once you know the degree, the leading coefficient and the zeros, you can sketch almost any polynomial by hand.