Functions and Their Evaluation

Evaluating Functions at Specific Values

Evaluating a function involves substituting a given value for the variable and simplifying the result.

• Function Notation: If is a function, then means the value of the function when .

• Example: For , find , , , and by substituting each value into the function.

Graphing and Analyzing Functions

Graphing Absolute Value and Square Root Functions

Understanding the shape and domain/range of basic functions is essential in precalculus.

• Absolute Value Function: is a V-shaped graph opening downward, with vertex at .

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

• Domain: The set of all possible input values () for which the function is defined.

• Range: The set of all possible output values ().

• Example: Graph and estimate its domain and range.

Identifying Relative (Local) Maxima and Minima

Relative maxima and minima are points where a function reaches a highest or lowest value locally.

• Relative Maximum: A point where the function changes from increasing to decreasing.

• Relative Minimum: A point where the function changes from decreasing to increasing.

• Example: By examining a graph, identify the -values where the function has local maxima or minima.

Intervals of Increase, Decrease, and Constancy

Functions can be increasing, decreasing, or constant over different intervals.

• Increasing: for in the interval.

• Decreasing: for in the interval.

• Constant: for all in the interval.

• Example: From a graph, state the intervals where the function is increasing, decreasing, or constant.

Algebraic Techniques

Solving Linear and Quadratic Equations

Solving equations is a fundamental skill in precalculus, including both linear and quadratic types.

• Linear Equations: Equations of the form .

• Quadratic Equations: Equations of the form .

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

Solving Equations Involving Absolute Value

Absolute value equations may have two solutions, depending on the sign of the expression inside the absolute value.

• General Form: leads to or .

• Inequalities: or are solved by considering both cases and graphing the solution on a number line.

• Example: Solve and .

Solving Radical Equations

Equations involving square roots require isolating the radical and then squaring both sides to eliminate it.

• Example: Solve by isolating one radical and squaring both sides.

Function Operations and Composition

Function Composition

Composing functions involves substituting one function into another.

• Notation: .

• Domain of Composition: The set of values for which is in the domain of .

• Example: Given and , find and its domain.

Complex Numbers

Operations with Complex Numbers

Complex numbers are numbers of the form , where .

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

• Multiplication: Use distributive property and .

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

• Example: Simplify and .

Linear Equations and Systems

Classifying Pairs of Lines

Two lines in the plane can be parallel, perpendicular, or neither, depending on their slopes.

• Parallel Lines: Have equal slopes.

• Perpendicular Lines: Slopes are negative reciprocals.

• Example: Determine if the lines and are parallel, perpendicular, or neither.

Word Problems Involving Systems of Equations

Many real-world problems can be modeled and solved using systems of linear equations.

• Example: If Martin has $2000 invested in two accounts with different interest rates, set up a system of equations to find how much is in each account.

Summary Table: Types of Equations and Solution Methods