Functions and Their Graphs

Function Basics

Functions are fundamental objects in precalculus, describing relationships between variables. Understanding their properties and how to represent them graphically is essential.

• Definition: A function is a rule that assigns to each element in a set called the domain exactly one element in a set called the range.

• Function Notation: If is a function, denotes the output when is the input.

• Domain and Range: The domain is the set of all possible inputs; the range is the set of all possible outputs.

• Composite Functions: The composition means applying first, then .

• Inverse Functions: If is invertible, reverses the action of $f$.

• Difference Quotient: Used to measure the average rate of change: $

• Properties: Functions can be increasing, decreasing, have maximum or minimum values.

Graphing Functions

• Transformations: Shifting, stretching, compressing, and reflecting graphs to obtain new functions.

• Sketching: Identify key features such as intercepts, asymptotes, and symmetry.

• Example: The graph of is a parabola shifted right by 2 and up by 3.

Polynomial and Rational Functions

Polynomial Functions

Polynomial functions are sums of powers of with constant coefficients. Their graphs and zeros are central to precalculus.

• General Form:

• Zeros: Solutions to are called roots or zeros.

• Vertex: For quadratics , the vertex is at .

• Sketching: Identify end behavior, intercepts, and turning points.

Rational Functions

• Definition: A rational function is a ratio of two polynomials: .

• Vertical Asymptotes: Occur where and .

• Zeros: Occur where and .

• Example: has a vertical asymptote at and zeros at .

Radicals and Rational Expressions

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. Simplifying and solving equations with radicals is a key skill.

• Definition: denotes the nth root of .

• Simplifying: Combine like terms and rationalize denominators when necessary.

• Solving Radical Equations: Isolate the radical, then raise both sides to the appropriate power.

• Example: Solve :

  • Square both sides:

  • So

Rational Expressions

• Definition: An expression of the form where .

• Simplifying: Factor numerator and denominator, then reduce common factors.

• Operations: Addition, subtraction, multiplication, and division follow the rules for fractions.

Complex Numbers

Definition and Operations

Complex numbers extend the real numbers to include solutions to equations like .

• Definition: , where and are real numbers and is the imaginary unit, .

• Operations: Add, subtract, multiply, and divide using algebraic rules and .

• Example:

Exponential and Logarithmic Functions

Exponential Functions

Exponential functions model growth and decay processes and have the form .

• Properties: , ; the function is always positive.

• Graphing: Exponential growth if , decay if .

• Example: grows rapidly as increases.

Logarithmic Functions

• Definition: is the inverse of .

• Properties: ; .

• Solving Equations: Use properties of logarithms to combine or expand expressions, then solve for the variable.

• Example: Solve :

  • Take of both sides:

Equations and Inequalities

Quadratic, Polynomial, Rational, Radical, and Absolute Value Equations

Solving various types of equations is a core skill in precalculus.

• Quadratic Equations: can be solved by factoring, completing the square, or the quadratic formula: $

• Polynomial Equations: Set the polynomial equal to zero and factor or use synthetic division.

• Rational Equations: Multiply both sides by the least common denominator to clear fractions.

• Radical Equations: Isolate the radical and raise both sides to a power to eliminate it.

• Absolute Value Equations: has solutions or .

• Inequalities: Solve similarly to equations, but consider the direction of the inequality when multiplying or dividing by negatives.

Applications and Technology

Applications

Precalculus concepts are used to solve real-world problems, such as modeling population growth, financial calculations, and physics problems.

• Example: Exponential growth models, such as for population growth.

• Quadratic Applications: Projectile motion, area problems, and optimization.

Technology

• Graphing Calculators: Used to plot functions, find zeros, and analyze graphs.

• Software: Tools like Desmos or GeoGebra enhance understanding of function behavior.