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Study Guides

MTH 163, Precalculus

Functions and Function Notation

Use arrow notation
Introduction to Functions and Function Notation
Solve applied problems involving rational functions
Determine whether a relation represents a function
Find the domains of rational functions
Find the input and output values of a function
Identify vertical asymptotes
Determine whether a function is one-to-one
Identify horizontal asymptotes
Use the vertical line test to identify functions
Graph rational functions
Graph the functions listed in the library of functions
Key Concepts & Glossary
Key Concepts & Glossary
Section Exercises
Section Exercises
Solutions
Solutions

Inverses and Radical Functions

Domain and Range

Introduction to Inverses and Radical Functions
Introduction to Domain and Range
Find the inverse of a polynomial function
Find the domain of a function defined by an equation
Restrict the domain to find the inverse of a polynomial function
Use notations to specify domain and range
Key Concepts & Glossary
Find domain and range from graphs
Section Exercises
Find domains and ranges of the toolkit functions
Solutions
Graph piecewise-defined functions

Modeling Using Variation

Key Concepts & Glossary
Introduction to Modeling Using Variation
Section Exercises
Solve direct variation problems
Solutions
Solve inverse variation problems

Rates of Change and Behavior of Graphs

Solve problems involving joint variation
Introduction to Rates of Change and Behaviors of Graphs
Key Concepts & Glossary
Find the average rate of change of a function
Section Exercises
Use a graph to determine where a function is increasing, decreasing, or constant
Solutions
Use a graph to locate the absolute maximum and absolute minimum

Exponential Functions

Key Concepts & Glossary
Introduction to Exponential Functions
Section Exercises
Evaluate exponential functions
Solutions
Find the equation of an exponential function

Composition of Functions

Use compound interest formulas
Introduction to Composition of Functions
Evaluate exponential functions with base e
Combine functions using algebraic operations
Key Concepts & Glossary
Create a new function by composition of functions
Section Exercises
Evaluate composite functions
Solutions
Find the domain of a composite function

Graphs of Exponential Functions

Decompose a composite function into its component functions
Introduction to Graphs of Exponential Functions
Key Concepts & Glossary
Graph exponential functions
Section Exercises
Graph exponential functions using transformations
Solutions
Key Concepts

Transformation of Functions

Section Exercises
Introduction to Transformation of Functions
Solutions
Graph functions using vertical and horizontal shifts

Logarithmic Functions

Graph functions using reflections about the x-axis and the y-axis
Introduction to Logarithmic Functions
Determine whether a function is even, odd, or neither from its graph
Convert from logarithmic to exponential form
Graph functions using compressions and stretches
Convert from exponential to logarithmic form
Combine vertical and horizontal shifts
Evaluate logarithms
Key Concepts & Glossary
Use common logarithms
Section Exercises
Use natural logarithms
Solutions
Key Concepts & Glossary

Absolute Value Functions

Section Exercises
Introduction to Absolute Value Functions
Solutions
Graph an absolute value function

Graphs of Logarithmic Functions

Solve an absolute value equation
Introduction to Graphs of Logarithmic Functions
Solve an absolute value inequality
Identify the domain of a logarithmic function
Key Concepts & Glossary
Graph logarithmic functions
Section Exercises
Graphing Transformations of Logarithmic Functions
Solutions
Key Concepts

Inverse Functions

Section Exercises
Introduction to Inverse Functions
Solutions
Verify inverse functions

Logarithmic Properties

Determine the domain and range of an inverse function
Introduction to Logarithmic Properties
Find or evaluate the inverse of a function
Use the product rule for logarithms
Use the graph of a function to graph its inverse
Use the quotient and power rules for logarithms
Key Concepts & Glossary
Expand logarithmic expressions
Section Exercises
Condense logarithmic expressions
Solutions
Use the change-of-base formula for logarithms

Linear Functions

Key Terms & Glossary
Introduction to Linear Functions
Section Exercises
Represent a linear function
Solutions
Determine whether a linear function is increasing, decreasing, or constant

Exponential and Logarithmic Equations

Calculate and interpret slope
Introduction to Exponential and Logarithmic Equations
Write the point-slope form of an equation
Use like bases to solve exponential equations
Write and interpret a linear function
Use logarithms to solve exponential equations
Key Concepts & Glossary
Use the definition of a logarithm to solve logarithmic equations
Section Exercises
Use the one-to-one property of logarithms to solve logarithmic equations
Solutions
Solve applied problems involving exponential and logarithmic equations

Graphs of Linear Functions

Key Concepts & Glossary
Introduction to Graphs of Linear Functions
Section Exercises
Graph linear functions
Solutions
Write the equation for a linear function from the graph of a line

Exponential and Logarithmic Models

Given the equations of two lines, determine whether their graphs are parallel or perpendicular
Introduction to Exponential and Logarithmic Models
Write the equation of a line parallel or perpendicular to a given line
Model exponential growth and decay
Solve a system of linear equations
Use Newton’s Law of Cooling
Key Concepts & Glossary
Use logistic-growth models
Section Exercises
Choose an appropriate model for data
Solutions
Key Concepts & Glossary

Modeling with Linear Functions

Section Exercises
Introduction to Modeling With Linear Functions
Solutions
Identify steps for modeling and solving

Fitting Exponential Models to Data

Build linear models
Introduction to Fitting Exponential Models to Data
Build systems of linear models
Build an exponential model from data
Section Exercises
Build a logarithmic model from data
Solutions
Build a logistic model from data

Fitting Linear Models to Data

Key Concepts
Introduction to Fitting Linear Models to Data
Section Exercises
Draw and interpret scatter plots
Solutions
Find the line of best fit

Systems of Linear Equations: Two Variables

Distinguish between linear and nonlinear relations
Introduction to Systems of Linear Equations: Two Variables
Use a linear model to make predictions
Solving Systems of Equations by Graphing
Key Concepts & Glossary
Solving Systems of Equations by Substitution
Section Exercises
Solving Systems of Equations in Two Variables by the Addition Method
Solutions
Identifying and Expressing Solutions to Systems of Equations

Complex Numbers

Using Systems of Equations to Investigate Profits
Introduction to Complex Numbers
Key Concepts & Glossary
Express square roots of negative numbers as multiples of  i
Section Exercises
Plot complex numbers on the complex plane
Solutions
Add and subtract complex numbers

Systems of Linear Equations: Three Variables

Multiply and divide complex numbers
Introduction to Systems of Linear Equations: Three Variables
Key Concepts & Glossary
Solving Systems of Three Equations in Three Variables
Section Exercises
Inconsistent and Dependent Systems in Three Variables
Solutions
Key Concepts & Glossary

Quadratic Functions

Section Exercises
Introduction to Quadratic Functions
Solutions
Recognize characteristics of parabolas

Systems of Nonlinear Equations and Inequalities: Two Variables

Understand how the graph of a parabola is related to its quadratic function
Introduction to Systems of Nonlinear Equations and Inequalities: Two Variables
Determine a quadratic function’s minimum or maximum value
Solving a System of Nonlinear Equations Using Substitution
Solve problems involving a quadratic function’s minimum or maximum value
Solving a System of Nonlinear Equations Using Elimination
Key Concepts & Glossary
Graphing Nonlinear Inequalities and Systems of Nonlinear Inequalities
Section Exercises
Key Concepts & Glossary
Solutions
Section Exercises

Power Functions and Polynomial Functions

Solutions
Introduction to Power Functions and Polynomial Functions

Matrices and Matrix Operations

Identify power functions
Introduction: Matrices and Matrix Operations
Identify end behavior of power functions
Finding the Sum and Difference of Two Matrices
Identify polynomial functions
Finding Scalar Multiples of a Matrix
Identify the degree and leading coefficient of polynomial functions
Finding the Product of Two Matrices
Key Concepts & Glossary
Key Concepts & Glossary
Section Exercises
Section Exercises
Solutions
Solutions

Graphs of Polynomial Functions

Solving Systems with Gaussian Elimination

Introduction to Graphs of Polynomial Functions
Introduction to Solving Systems with Gaussian Elimination
Recognize characteristics of graphs of polynomial functions
The Augmented Matrix of a System of Equations
Use factoring to find zeros of polynomial functions
Performing Row Operations on a Matrix
Identify zeros and their multiplicities
Solving a System of Linear Equations Using Matrices
Determine end behavior
Key Concepts & Glossary
Understand the relationship between degree and turning points
Section Exercises
Graph polynomial functions
Solutions
Use the Intermediate Value Theorem

Solving Systems with Inverses

Key Concepts & Glossary
Introduction to Solving Systems with Inverses
Section Exercises
Finding the Inverse of a Matrix
Solutions
Solving a System of Linear Equations Using the Inverse of a Matrix

Dividing Polynomials

Key Concepts & Glossary
Introduction to Dividing Polynomials
Section Exercises
Use long division to divide polynomials
Solutions
Use synthetic division to divide polynomials

Solving Systems with Cramer's Rule

Use polynomial division to solve application problems
Introduction to Solving Systems with Cramer's Rule
Key Concepts & Glossary
Using Cramer’s Rule to Solve a System of Two Equations in Two Variables
Section Exercises
Using Cramer’s Rule to Solve a System of Three Equations in Three Variables
Solutions
Understanding Properties of Determinants

Zeros of Polynomial Functions

Key Concepts & Glossary
Introduction to Zeros of Polynomials
Solutions
Evaluate a polynomial using the Remainder Theorem**
Section Exercises
Use the Factor Theorem to solve a polynomial equation

Practice Tests

Use the Rational Zero Theorem to find rational zeros
Functions Practice Test
Find zeros of a polynomial function
Solutions: Functions Practice Test
Use the Fundamental Theorem of Algebra
Linear Functions Practice Test
Use the Linear Factorization Theorem to find polynomials with given zeros
Solutions: Linear Functions Practice Test
Use Descartes’ Rule of Signs
Polynomial and Rational Functions Practice Test
Solve real-world applications of polynomial equations
Solutions: Polynomial and Rational Functions Practice Test
Key Concepts & Glossary
Exponential and Logarithmic Functions Practice Test
Section Exercises
Solutions: Exponential and Logarithmic Functions Practice Test
Solutions

Appendix

Rational Functions

Appendix
Introduction to Rational Functions