Functions and Their Evaluation

Evaluating Functions at Specific Points

Evaluating a function means substituting a given value for the variable and calculating the result. This is a foundational skill in precalculus, used to analyze and interpret mathematical models.

• Definition: For a function , the value is found by replacing with in the formula.

• Example: If , then .

• Application: Evaluating functions at various points helps determine their behavior and possible restrictions (domain).

Graphing Functions and Estimating Domain & Range

Absolute Value and Square Root Functions

Graphing functions visually represents their behavior, including domain (possible input values) and range (possible output values).

• Absolute Value Function: produces a 'V'-shaped graph shifted vertically.

• Square Root Function: is defined for and increases slowly as increases.

• Estimating Domain: The domain is the set of all for which the function is defined (e.g., for , ).

• Estimating Range: The range is the set of all possible output values.

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

Linear Equations and Relationships

Parallel and Perpendicular Lines

Understanding the relationship between lines is essential for analyzing geometric properties and solving systems.

• Parallel Lines: Two lines are parallel if their slopes are equal.

• Perpendicular Lines: Two lines are perpendicular if the product of their slopes is .

• Example: For and , both have slope $2$ and are parallel.

• Example: For and , slopes are and $2(-\frac{1}{2}) \times 2 = -1$, so they are perpendicular.

Word Problems: Systems of Equations

Investment and Interest Problems

Word problems often require setting up and solving systems of equations to find unknown quantities.

• Example: If $2000 is invested in two accounts, one at 2% and one at 9%, and $129 is earned in interest, set up equations to solve for the amount in each account.

• Formulas: where is interest, is principal, is rate, is time.

• Application: Translate the word problem into equations and solve using algebraic methods.

Graph Analysis: Maxima, Minima, and Intervals

Identifying Relative Maxima and Minima

Analyzing graphs allows us to determine where functions reach their highest or lowest points locally.

• Relative Maximum: A point where the function reaches a peak compared to nearby values.

• Relative Minimum: A point where the function reaches a trough compared to nearby values.

• Intervals of Increase/Decrease: The sections of the graph where the function is rising or falling.

• Example: By examining the graph, state intervals where the function increases, decreases, or remains constant.

Function Operations and Composition

Sum, Difference, Product, and Composition of Functions

Functions can be combined in various ways to create new functions.

• Sum:

• Difference:

• Product:

• Quotient:

• Composition:

• Domain of Composite Functions: The domain of is all in the domain of such that is in the domain of .

• Example: If and , then .

Complex Numbers and Simplification

Operations with Complex Numbers

Complex numbers are numbers of the form , where is the imaginary unit ().

• Addition/Subtraction: Combine real and imaginary parts separately.

• Multiplication: Use distributive property and .

• Example:

• Division: Multiply numerator and denominator by the conjugate of the denominator.

• Example:

Solving Equations: Quadratic, Radical, and Absolute Value

Quadratic Equations

Quadratic equations are equations of the form .

• Factoring: Express as a product of binomials and set each factor to zero.

• Quadratic Formula:

• Example: Solve by rearranging and factoring or using the quadratic formula.

Radical Equations

• Definition: Equations involving roots, such as .

• Solving: Isolate the radical, square both sides, and solve the resulting equation.

Absolute Value Equations and Inequalities

• Definition: is the distance from to $0$ on the number line.

• Equations: has solutions and .

• Inequalities: means .

• Example: Solve and graph the solution on a number line.

Summary Table: Types of Equations and Solution Methods

Additional info:

• Some examples and exercises are labeled as "You Try," indicating practice problems for students.

• Graphs are referenced for analysis of maxima, minima, and intervals of increase/decrease, which are key concepts in function analysis.

• Problems cover a broad range of precalculus topics, including function evaluation, graphing, algebraic manipulation, and solving equations.