Skip to Content
OpenStax Logo
Precalculus

Key Concepts

PrecalculusKey Concepts
  1. Preface
  2. 1 Functions
    1. Introduction to Functions
    2. 1.1 Functions and Function Notation
    3. 1.2 Domain and Range
    4. 1.3 Rates of Change and Behavior of Graphs
    5. 1.4 Composition of Functions
    6. 1.5 Transformation of Functions
    7. 1.6 Absolute Value Functions
    8. 1.7 Inverse Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  3. 2 Linear Functions
    1. Introduction to Linear Functions
    2. 2.1 Linear Functions
    3. 2.2 Graphs of Linear Functions
    4. 2.3 Modeling with Linear Functions
    5. 2.4 Fitting Linear Models to Data
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  4. 3 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 3.1 Complex Numbers
    3. 3.2 Quadratic Functions
    4. 3.3 Power Functions and Polynomial Functions
    5. 3.4 Graphs of Polynomial Functions
    6. 3.5 Dividing Polynomials
    7. 3.6 Zeros of Polynomial Functions
    8. 3.7 Rational Functions
    9. 3.8 Inverses and Radical Functions
    10. 3.9 Modeling Using Variation
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Review Exercises
    15. Practice Test
  5. 4 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 4.1 Exponential Functions
    3. 4.2 Graphs of Exponential Functions
    4. 4.3 Logarithmic Functions
    5. 4.4 Graphs of Logarithmic Functions
    6. 4.5 Logarithmic Properties
    7. 4.6 Exponential and Logarithmic Equations
    8. 4.7 Exponential and Logarithmic Models
    9. 4.8 Fitting Exponential Models to Data
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  6. 5 Trigonometric Functions
    1. Introduction to Trigonometric Functions
    2. 5.1 Angles
    3. 5.2 Unit Circle: Sine and Cosine Functions
    4. 5.3 The Other Trigonometric Functions
    5. 5.4 Right Triangle Trigonometry
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  7. 6 Periodic Functions
    1. Introduction to Periodic Functions
    2. 6.1 Graphs of the Sine and Cosine Functions
    3. 6.2 Graphs of the Other Trigonometric Functions
    4. 6.3 Inverse Trigonometric Functions
    5. Key Terms
    6. Key Equations
    7. Key Concepts
    8. Review Exercises
    9. Practice Test
  8. 7 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 7.1 Solving Trigonometric Equations with Identities
    3. 7.2 Sum and Difference Identities
    4. 7.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 7.4 Sum-to-Product and Product-to-Sum Formulas
    6. 7.5 Solving Trigonometric Equations
    7. 7.6 Modeling with Trigonometric Equations
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. Practice Test
  9. 8 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 8.1 Non-right Triangles: Law of Sines
    3. 8.2 Non-right Triangles: Law of Cosines
    4. 8.3 Polar Coordinates
    5. 8.4 Polar Coordinates: Graphs
    6. 8.5 Polar Form of Complex Numbers
    7. 8.6 Parametric Equations
    8. 8.7 Parametric Equations: Graphs
    9. 8.8 Vectors
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  10. 9 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 9.1 Systems of Linear Equations: Two Variables
    3. 9.2 Systems of Linear Equations: Three Variables
    4. 9.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 9.4 Partial Fractions
    6. 9.5 Matrices and Matrix Operations
    7. 9.6 Solving Systems with Gaussian Elimination
    8. 9.7 Solving Systems with Inverses
    9. 9.8 Solving Systems with Cramer's Rule
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  11. 10 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 10.1 The Ellipse
    3. 10.2 The Hyperbola
    4. 10.3 The Parabola
    5. 10.4 Rotation of Axes
    6. 10.5 Conic Sections in Polar Coordinates
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  12. 11 Sequences, Probability and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 11.1 Sequences and Their Notations
    3. 11.2 Arithmetic Sequences
    4. 11.3 Geometric Sequences
    5. 11.4 Series and Their Notations
    6. 11.5 Counting Principles
    7. 11.6 Binomial Theorem
    8. 11.7 Probability
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  13. 12 Introduction to Calculus
    1. Introduction to Calculus
    2. 12.1 Finding Limits: Numerical and Graphical Approaches
    3. 12.2 Finding Limits: Properties of Limits
    4. 12.3 Continuity
    5. 12.4 Derivatives
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  14. A | Basic Functions and Identities
  15. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  16. Index

1.1 Functions and Function Notation

  • A relation is a set of ordered pairs. A function is a specific type of relation in which each domain value, or input, leads to exactly one range value, or output. See Example 1 and Example 2.
  • Function notation is a shorthand method for relating the input to the output in the form y=f( x ). y=f( x ).See Example 3 and Example 4.
  • In tabular form, a function can be represented by rows or columns that relate to input and output values. See Example 5.
  • To evaluate a function, we determine an output value for a corresponding input value. Algebraic forms of a function can be evaluated by replacing the input variable with a given value. See Example 6 and Example 7.
  • To solve for a specific function value, we determine the input values that yield the specific output value. See Example 8.
  • An algebraic form of a function can be written from an equation. See Example 9 and Example 10.
  • Input and output values of a function can be identified from a table. See Example 11.
  • Relating input values to output values on a graph is another way to evaluate a function. See Example 12.
  • A function is one-to-one if each output value corresponds to only one input value. See Example 13.
  • A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point. See Example 14.
  • The graph of a one-to-one function passes the horizontal line test. See Example 15.

1.2 Domain and Range

  • The domain of a function includes all real input values that would not cause us to attempt an undefined mathematical operation, such as dividing by zero or taking the square root of a negative number.
  • The domain of a function can be determined by listing the input values of a set of ordered pairs. See Example 1.
  • The domain of a function can also be determined by identifying the input values of a function written as an equation. See Example 2, Example 3, and Example 4.
  • Interval values represented on a number line can be described using inequality notation, set-builder notation, and interval notation. See Example 5.
  • For many functions, the domain and range can be determined from a graph. See Example 6 and Example 7.
  • An understanding of toolkit functions can be used to find the domain and range of related functions. See Example 8, Example 9, and Example 10.
  • A piecewise function is described by more than one formula. See Example 11 and Example 12.
  • A piecewise function can be graphed using each algebraic formula on its assigned subdomain. See Example 13.

1.3 Rates of Change and Behavior of Graphs

  • A rate of change relates a change in an output quantity to a change in an input quantity. The average rate of change is determined using only the beginning and ending data. See Example 1.
  • Identifying points that mark the interval on a graph can be used to find the average rate of change. See Example 2.
  • Comparing pairs of input and output values in a table can also be used to find the average rate of change. See Example 3.
  • An average rate of change can also be computed by determining the function values at the endpoints of an interval described by a formula. See Example 4 and Example 5.
  • The average rate of change can sometimes be determined as an expression. See Example 6.
  • A function is increasing where its rate of change is positive and decreasing where its rate of change is negative. See Example 7.
  • A local maximum is where a function changes from increasing to decreasing and has an output value larger (more positive or less negative) than output values at neighboring input values.
  • A local minimum is where the function changes from decreasing to increasing (as the input increases) and has an output value smaller (more negative or less positive) than output values at neighboring input values.
  • Minima and maxima are also called extrema.
  • We can find local extrema from a graph. See Example 8 and Example 9.
  • The highest and lowest points on a graph indicate the maxima and minima. See Example 10.

1.4 Composition of Functions

  • We can perform algebraic operations on functions. See Example 1.
  • When functions are combined, the output of the first (inner) function becomes the input of the second (outer) function.
  • The function produced by combining two functions is a composite function. See Example 2 and Example 3.
  • The order of function composition must be considered when interpreting the meaning of composite functions. See Example 4.
  • A composite function can be evaluated by evaluating the inner function using the given input value and then evaluating the outer function taking as its input the output of the inner function.
  • A composite function can be evaluated from a table. See Example 5.
  • A composite function can be evaluated from a graph. See Example 6.
  • A composite function can be evaluated from a formula. See Example 7.
  • The domain of a composite function consists of those inputs in the domain of the inner function that correspond to outputs of the inner function that are in the domain of the outer function. See Example 8 and Example 9.
  • Just as functions can be combined to form a composite function, composite functions can be decomposed into simpler functions.
  • Functions can often be decomposed in more than one way. See Example 10.

1.5 Transformation of Functions

  • A function can be shifted vertically by adding a constant to the output. See Example 1 and Example 2.
  • A function can be shifted horizontally by adding a constant to the input. See Example 3, Example 4, and Example 5.
  • Relating the shift to the context of a problem makes it possible to compare and interpret vertical and horizontal shifts. See Example 6.
  • Vertical and horizontal shifts are often combined. See Example 7 and Example 8.
  • A vertical reflection reflects a graph about the x- x-axis. A graph can be reflected vertically by multiplying the output by –1.
  • A horizontal reflection reflects a graph about the y- y-axis. A graph can be reflected horizontally by multiplying the input by –1.
  • A graph can be reflected both vertically and horizontally. The order in which the reflections are applied does not affect the final graph. See Example 9.
  • A function presented in tabular form can also be reflected by multiplying the values in the input and output rows or columns accordingly. See Example 10.
  • A function presented as an equation can be reflected by applying transformations one at a time. See Example 11.
  • Even functions are symmetric about the y- y-axis, whereas odd functions are symmetric about the origin.
  • Even functions satisfy the condition f(x)=f(x). f(x)=f(x).
  • Odd functions satisfy the condition f(x)=f(x). f(x)=f(x).
  • A function can be odd, even, or neither. See Example 12.
  • A function can be compressed or stretched vertically by multiplying the output by a constant. See Example 13, Example 14, and Example 15.
  • A function can be compressed or stretched horizontally by multiplying the input by a constant. See Example 16, Example 17, and Example 18.
  • The order in which different transformations are applied does affect the final function. Both vertical and horizontal transformations must be applied in the order given. However, a vertical transformation may be combined with a horizontal transformation in any order. See Example 19 and Example 20.

1.6 Absolute Value Functions

  • The absolute value function is commonly used to measure distances between points. See Example 1.
  • Applied problems, such as ranges of possible values, can also be solved using the absolute value function. See Example 2.
  • The graph of the absolute value function resembles a letter V. It has a corner point at which the graph changes direction. See Example 3.
  • In an absolute value equation, an unknown variable is the input of an absolute value function.
  • If the absolute value of an expression is set equal to a positive number, expect two solutions for the unknown variable. See Example 4.
  • An absolute value equation may have one solution, two solutions, or no solutions. See Example 5.
  • An absolute value inequality is similar to an absolute value equation but takes the form | A |<B,| A |B,| A |>B,or | A |B. | A |<B,| A |B,| A |>B,or | A |B.It can be solved by determining the boundaries of the solution set and then testing which segments are in the set. See Example 6.
  • Absolute value inequalities can also be solved graphically. See Example 7.

1.7 Inverse Functions

  • If g(x) g(x)is the inverse of f(x), f(x),then g(f(x))=f(g(x))=x. g(f(x))=f(g(x))=x.See Example 1, Example 2, and Example 3.
  • Each of the toolkit functions has an inverse. See Example 4.
  • For a function to have an inverse, it must be one-to-one (pass the horizontal line test).
  • A function that is not one-to-one over its entire domain may be one-to-one on part of its domain.
  • For a tabular function, exchange the input and output rows to obtain the inverse. See Example 5.
  • The inverse of a function can be determined at specific points on its graph. See Example 6.
  • To find the inverse of a formula, solve the equation y=f(x) y=f(x)for x xas a function of y. y.Then exchange the labels x xand y. y.See Example 7, Example 8, and Example 9.
  • The graph of an inverse function is the reflection of the graph of the original function across the line y=x. y=x.See Example 10.
Citation/Attribution

Want to cite, share, or modify this book? This book is Creative Commons Attribution License 4.0 and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/precalculus/pages/1-introduction-to-functions
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/precalculus/pages/1-introduction-to-functions
Citation information

© Feb 10, 2020 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License 4.0 license. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.