Functions

Introduction to Functions, Domain, and Range

Functions are fundamental objects in algebra that describe relationships between sets. Understanding their domain and range is essential for analyzing their behavior.

• Function: A rule that assigns each element in a set (domain) to exactly one element in another set (range).

• Domain: The set of all possible input values (x-values) for the function.

• Range: The set of all possible output values (y-values) produced by the function.

• Example: For , the domain is all real numbers, and the range is all non-negative real numbers.

Function Behaviors and Properties

Functions can be classified and analyzed based on their behaviors, such as increasing, decreasing, and constant intervals.

• Increasing Function: A function is increasing on an interval if, as x increases, f(x) increases.

• Decreasing Function: A function is decreasing on an interval if, as x increases, f(x) decreases.

• Constant Function: A function is constant on an interval if f(x) remains the same as x changes.

• Example: is increasing everywhere.

Linear vs. Non-Linear Functions and Transformations

Linear functions have constant rates of change, while non-linear functions do not. Transformations modify functions by shifting, stretching, or reflecting their graphs.

• Linear Function: Has the form .

• Non-Linear Function: Examples include quadratic, cubic, exponential, etc.

• Transformations: Include translations, reflections, stretches, and compressions.

• Example: shifted up by 3 units: .

Variations

Variation describes how one quantity changes in relation to another. Common types include direct, inverse, and joint variation.

• Direct Variation:

• Inverse Variation:

• Joint Variation:

• Example: If varies directly with , and , then .

Linear and Quadratic Functions

Linear Functions

Linear functions model constant rates of change and are represented by straight lines.

• General Form:

• Slope (m): Measures the rate of change.

• Y-intercept (b): The value of when .

• Example:

Quadratic Functions and Their Zeros

Quadratic functions are polynomials of degree 2 and their zeros are the solutions to .

• Standard Form:

• Zeros: Values of where

• Quadratic Formula:

• Example: has zeros at and .

Quadratic Function Properties and Modeling

Quadratic functions have parabolic graphs and are used in modeling various real-world phenomena.

• Vertex: The highest or lowest point of the parabola, given by

• Axis of Symmetry: Vertical line

• Example: Projectile motion modeled by

Complex Zeros and Absolute Value Equations/Inequalities

Quadratic equations may have complex zeros if the discriminant is negative. Absolute value equations and inequalities involve expressions within absolute value bars.

• Complex Zeros: Occur when

• Absolute Value Equation: has solutions or

• Absolute Value Inequality: implies

• Example: has zeros and

Systems of Equations

Systems of Linear and Non-Linear Equations

Systems of equations involve solving for variables that satisfy multiple equations simultaneously.

• Linear System: All equations are linear.

• Non-Linear System: At least one equation is non-linear.

• Methods: Substitution, elimination, and graphical methods.

• Example: Solve and .

Polynomial and Rational Functions

Polynomial and Power Functions

Polynomial functions are sums of powers of x with coefficients. Power functions have the form .

• Polynomial Function:

• Power Function:

• Example:

Rational Functions and Their Attributes

Rational functions are ratios of polynomials. Their attributes include domain, asymptotes, and intercepts.

• Rational Function: where

• Vertical Asymptote: Occurs where

• Horizontal Asymptote: Determined by degrees of and

• Example: has a vertical asymptote at

Rational Functions Analysis

Analyzing rational functions involves finding intercepts, asymptotes, and behavior near undefined points.

• Find Domain: Exclude values where denominator is zero.

• Find Asymptotes: Set denominator to zero for vertical, compare degrees for horizontal.

• Example:

Composite, Inverse, Exponential, and Logarithmic Functions

Composite Functions

Composite functions combine two functions, applying one after the other.

• Notation:

• Example: If and , then

Invertibility and Inverse Functions

A function is invertible if each output corresponds to exactly one input. The inverse function reverses the original mapping.

• Inverse Function: satisfies

• Test for Invertibility: Function must be one-to-one (pass the horizontal line test).

• Example: has inverse

Exponential Functions

Exponential functions have the form , where and .

• Growth: If , function increases rapidly.

• Decay: If , function decreases rapidly.

• Example:

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions.

• General Form:

• Properties: and

• Example:

Applications of Exponential and Logarithmic Functions

These functions are used in modeling growth, decay, and other real-world phenomena.

• Exponential Growth:

• Exponential Decay:

• Logarithmic Applications: pH in chemistry, Richter scale for earthquakes

• Example: Population growth modeled by

Review and Exam Preparation

Review Sessions

Review sessions are scheduled before exams to reinforce concepts and practice problem-solving.

• Key Strategies: Practice with sample problems, review formulas, and clarify concepts.

• Example: Review quadratic formula and its applications.

Exam Days

Exams assess understanding of course material, including functions, equations, and applications.

• Preparation: Review all major topics, practice solving equations, and understand function properties.

• Example: Exam may include solving systems of equations and analyzing function graphs.