Precalculus Algebra

Sets
Notations for Sets
Set Membership and Set Inclusion
2 Topics
Set Membership
Set Inclusion and Subsets
Set Operations
4 Topics
Set Unions
Set Intersections
Set Complements and Set Differences
Combining Set Operations
Exponents and Exponential Expressions
What Exponents Actually Mean
Arithmetical Rules For Exponential Expressions
2 Topics
Exponent Rules
Negative Exponents and Cancellation
Polynomial Arithmetic
What Is (and Isn’t) a Polynomial
Adding and Subtracting Polynomials
Multiplying Polynomials (aka “Distribution” or “FOILing”)
Factoring
Basic Factoring Methods
2 Topics
Factoring Out Greatest Common Factors
Factoring By Grouping
Factoring Quadratic Expressions
2 Topics
Factoring Quadratics where \(a=1\)
Factoring Harder Quadratics, where a isn’t 1
Factoring Differences of Squares
Rational Expressions
Arithmetic with Rational Expressions
3 Topics
Multiplying and Dividing Rational Expressions, and Cancellation
Adding, Subtracting, and Common Denominator Finding with Rational Expressions
Simplifying Compound/Complex Fractions
Radical Expressions
Simplifying Radical Expressions
4 Topics
Radical Cancellation and Fractional Power Representation
The Multiplication Rule for Radicals and Factor Trees
Simplifying Radical Expressions Involving Multiplication and Division
Adding, Subtracting, and FOILing with Radical Expressions
Rationalizing Denominators
2 Topics
Basic Rationalizing of Denominators
Rationalizing Denominators Using Conjugates
Functions Basics
What Functions Are and How They are Represented
4 Topics
Using Potato Diagrams
Using Sets/Lists of Ordered Pairs and Tables
Using a Given (Algebraic) Rule
Using Graphs and Plotting Them from Rules
Domains and Ranges of Functions
2 Topics
Finding Domains and Ranges Using Potato Diagrams and Sets of Ordered Pairs or Tables
Finding Domains and Ranges Using Graphs of Functions
Function Arithmetic (with Numerical Substitutions)
3 Topics
Basic Function Arithmetic Using Function Rules, and Numerical Substitutions
Function Arithmetic: Using Tables to Evaluate Various Combinations of Functions at a Given Value
Function Arithmetic: Using Graphs to Evaluate Various Combinations of Functions at a Given Value
Function Composition: Plugging One Function Into Another
3 Topics
Composing Two Function Rules
Composing Functions Using Tables
Composing Functions Using Graphs
The Difference Quotient
New Functions Constructed From Old
A Library of Commonly Used Functions (and Their Graphs)
Function Transformations: Shifting, Scaling, Reflecting
Piecewise-Defined Functions
2 Topics
Evaluating Piecewise-Defined Functions at a Given x-Value
Graphing Piecewise-Defined Functions
Linear Functions
Intro to Linear Functions and their Uses
Graphing Linear Functions from Slope-Intercept Form
Finding Equations of Lines Using Point-Slope Form
Average Rates of Change
Solving Linear Equations
Quadratic Functions
Solving Quadratic Equations
3 Topics
Using the Zero-Factor Property
Using the Square Root Property
Using the Quadratic Formula
Quadratic Functions Praxis
Exponential Functions
Exponential Functions Basics
2 Topics
Evaluating Exponential Functions at a Point
Graphing Exponential Functions and their Transformations
Compound Interest and the Natural Exponential Function
2 Topics
Compound Interest – Multiplying Your Money
The Natural Exponential Function
Exponential Functions Praxis
Logarithms
Logarithms
4 Topics
Evaluating Logarithmic Functions at Given x-Values
Graphing and Transforming Logarithmic Functions
Log Laws and Change-of-Base Formula
Solving Logarithmic and Exponential Equations
Logarithms Praxis
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Function Arithmetic (with Numerical Substitutions)

Precalculus Algebra Function Arithmetic (with Numerical Substitutions)

In What Follows…

We will discuss one of the key advantages of functions: the ability to build new functions from combinations of other functions!

Functions can be combined via the usual operations of addition, subtraction, multiplication, and division. They can be substituted into algebraic expressions. Functions can also be composed, which is essentially feeding the output of one function into another function as input. In the next few topics, we will cover the basic operations that can be performed on functions and will reserve future lessons for stuff about function composition.

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