Solving Inequalities and Equations

Linear and Quadratic Inequalities

Solving inequalities is a fundamental skill in precalculus, involving finding all values of the variable that make the inequality true.

• Linear Inequality: An inequality involving a linear expression, such as .

• Quadratic Inequality: An inequality involving a quadratic expression, such as or .

• Solution Method: Solve as you would an equation, but pay attention to the direction of the inequality when multiplying or dividing by negative numbers. For quadratics, find critical points and test intervals.

• Example: Solve .

  • First, isolate in both inequalities:

  • gives so

  • gives so

  • Combined:

Functions: Domain, Range, and Graphs

Finding the Domain

The domain of a function is the set of all input values (x-values) for which the function is defined.

• For square root functions, the expression under the root must be non-negative.

• For rational functions, the denominator cannot be zero.

• Example: Find the domain of .

  • Set

  • Domain:

Evaluating and Analyzing Functions

• Function Notation: denotes the output of function for input .

• Zeros of a Function: Values of for which .

• Y-intercept: The point where the graph crosses the y-axis ().

• X-intercepts: Points where the graph crosses the x-axis ().

• Example: Given :

  • Domain: All real numbers ()

  • Y-intercept:

  • X-intercepts: Solve

Graphical Analysis of Functions

Interpreting Graphs

Understanding the behavior of functions from their graphs is essential in precalculus.

• Increasing/Decreasing Intervals: Where the function rises or falls as increases.

• Local Minimum/Maximum: Points where the function reaches a lowest/highest value locally.

• Absolute Minimum/Maximum: The lowest/highest value of the function on a given interval.

• Example: Given a graph, list intervals where is decreasing, and identify local minima.

Transformations of Functions

Shifts and Reflections

Transformations change the position or shape of a graph.

• Vertical Shift: shifts the graph up by units.

• Horizontal Shift: shifts the graph right by units.

• Reflection: reflects the graph across the x-axis.

• Example: shifted left 2 units and up 3 units becomes .

Function Operations and Composition

Sum, Difference, Product, and Quotient

• Sum:

• Difference:

• Product:

• Quotient: ,

• Composition:

• Example: If and , then

Difference Quotient

The difference quotient is used to compute the average rate of change of a function and is foundational for calculus.

• Formula: ,

• Example: For , compute .

Special Types of Functions

Even and Odd Functions

• Even Function: for all in the domain. Graph is symmetric about the y-axis.

• Odd Function: for all in the domain. Graph is symmetric about the origin.

• Example: is odd; is even.

Linear Functions and Slope

Finding the Equation of a Line

• Slope-Intercept Form:

• Point-Slope Form:

• Perpendicular Lines: Slopes are negative reciprocals:

• Example: Find the equation of a line perpendicular to through .

  • Original slope: , so perpendicular slope:

  • Equation:

Table: Types of Function Transformations

Additional info:

• Some problems reference textbook exercises and encourage review of class notes and handouts, indicating these are practice and review questions for a Precalculus course.

• Graphical analysis includes identifying intervals of increase/decrease, local extrema, and absolute extrema, which are key skills in function analysis.

• Difference quotient and function composition are foundational for calculus readiness.