Precalculus Fundamentals: Study Guide and Practice Problems

Interval Notation

Understanding Interval Notation

Interval notation is a concise way to describe sets of real numbers, especially when expressing solutions to inequalities or domains of functions.

• Open Interval (a, b): All real numbers x such that a < x < b. Endpoints are not included.

• Closed Interval [a, b]: All real numbers x such that a ≤ x ≤ b. Endpoints are included.

• Half-Open Intervals: (a, b] means a < x ≤ b; [a, b) means a ≤ x < b.

• Infinite Intervals: Use (a, ∞) for x > a, [a, ∞) for x ≥ a, (-∞, b) for x < b, and (-∞, b] for x ≤ b.

• Union (∪): Combines two intervals ("or").

• Intersection (∩): Overlap of two intervals ("and").

Rules:

• Always use ( ) with ∞ or -∞ (they are not real numbers).

• Use [ ] when the endpoint is included (≤ or ≥).

• Use ( ) when the endpoint is not included (< or >).

Example: The set {x | -3 < x ≤ 5} is written as (-3, 5].

Table: Interval Notation, Set-Builder, and Graphical Representation

Interval Notation	Set-Builder	Graph Description
(a, b)	{x | a < x < b}	Open dots at a and b
[a, b]	{x | a ≤ x ≤ b}	Closed dots at a and b
(a, b]	{x | a < x ≤ b}	Open at a, closed at b
[a, b)	{x | a ≤ x < b}	Closed at a, open at b
(a, ∞)	{x | x > a}	Arrow right from open dot
[a, ∞)	{x | x ≥ a}	Arrow right from closed dot
(-∞, b)	{x | x < b}	Arrow left to open dot
(-∞, b]	{x | x ≤ b}	Arrow left to closed dot
(-∞, ∞)	ℝ	Entire number line

Simplifying Expressions with Roots

Properties of Radicals

Radical expressions can be simplified using several key properties:

• Product Rule: $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$ (for a, b ≥ 0)

• Quotient Rule: $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ (for a ≥ 0, b > 0)

• Even Roots: $\sqrt[n]{a^n} = |a|$ if n is even; $\sqrt[n]{a^n} = a$ if n is odd

Simplifying Steps:

• Factor out perfect squares (or cubes, etc.) from under the radical.

• Rationalize denominators: Multiply numerator and denominator by a suitable radical to eliminate radicals from the denominator.

Example: $\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$

Rationalizing Denominators

• For a single radical: $\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

• For binomials: Multiply by the conjugate. $\frac{1}{\sqrt{3} + \sqrt{2}} \cdot \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} - \sqrt{2}} = \sqrt{3} - \sqrt{2}$

Working with Inequalities

Solving Linear and Compound Inequalities

Solving inequalities is similar to solving equations, with one key exception: Multiplying or dividing both sides by a negative number reverses the inequality sign.

• Compound "AND" Inequality: $a < x < b$ means x is between a and b.

• Compound "OR" Inequality: $x < a$ or $x > b$ means x is outside the interval [a, b].

Absolute Value Inequalities

• $|X| < c$ means $-c < X < c$ (one interval, AND)

• $|X| > c$ means $X < -c$ or $X > c$ (two intervals, OR)

Example: $|2x-5| \leq 7$ becomes $-7 \leq 2x-5 \leq 7$

Fractional (Rational) Exponents

Definition and Properties

• $a^{1/n} = \sqrt[n]{a}$

• $a^{m/n} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$

• Exponent rules apply: $a^m \cdot a^n = a^{m+n}$, $a^m / a^n = a^{m-n}$, $(a^m)^n = a^{mn}$

• Negative exponents: $a^{-n} = 1/a^n$

Restrictions:

• If n is even, a must be ≥ 0.

• If n is odd, a can be any real number.

• Denominator cannot be zero.

Example: $8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4$

Determining if an Equation Represents a Function

Definition and Tests

• Function: Each input (x) has exactly one output (y).

• Vertical Line Test: If any vertical line crosses the graph more than once, it is not a function.

• Algebraic Check: If solving for y gives y = ±..., then for some x there are two y-values, so it is not a function.

Example: $x^2 + y^2 = 1$ is not a function (y = ±√(1-x²)).

Finding Domain and Range

Domain

• Division by zero: Exclude x-values that make the denominator zero.

• Even roots: The radicand (expression under the root) must be ≥ 0.

• Odd roots: Defined for all real numbers.

Example: $f(x) = \frac{1}{x-3}$ has domain $(-\infty, 3) \cup (3, \infty)$.

Range

• For linear functions: Range is $(-\infty, \infty)$.

• For quadratics: Find the vertex; range is $[min, \infty)$ or $(-\infty, max]$ depending on the direction.

• For square roots: Range is $[0, \infty)$.

Deciding Whether a Relation is a Function

From Ordered Pairs, Tables, or Mapping Diagrams

• A relation is a function if no x-value is repeated with different y-values.

• Check for repeated x-values with different outputs.

Example: {(-2, 3), (0, 1), (-2, 5)} is not a function (x = -2 has two outputs).

Deciding Whether a Function is One-to-One

Definition and Tests

• One-to-one: Each output corresponds to exactly one input.

• Horizontal Line Test: If any horizontal line crosses the graph more than once, the function is not one-to-one.

• From ordered pairs: No y-value is repeated for different x-values.

Example: $f(x) = x^2$ is not one-to-one (f(2) = f(-2) = 4).

Quick Reference Tables

Domain Restrictions Table

Feature	Restriction	How to Find
Denominator	≠ 0	Set denominator = 0, exclude those x-values
Even root	Radicand ≥ 0	Set radicand ≥ 0, solve
Odd root	None	Domain is $(-\infty, \infty)$
Logarithm	Argument > 0	Set argument > 0, solve

Absolute Value Inequality Patterns

Form	Meaning	Pattern
|X| < c	-c < X < c	One interval (AND)
|X| > c	X < -c OR X > c	Two intervals (OR)

Summary of Key Concepts

• Function: Each input has one output. Use the vertical line test or check for repeated x-values with different y-values.

• One-to-One: Each output has one input. Use the horizontal line test or check for repeated y-values with different x-values.

• Domain: Exclude values that cause division by zero or even roots of negatives.

• Interval Notation: Use ( ) for endpoints not included, [ ] for included, and always ( ) with ∞ or -∞.

• Radicals and Exponents: Apply exponent rules and rationalize denominators as needed.

• Inequalities: Flip the sign when multiplying/dividing by a negative; use interval notation for solutions.

Additional info: This guide covers foundational precalculus topics including interval notation, radicals, inequalities, exponents, functions, domain and range, and one-to-one functions, with examples and tables for quick reference.