MAT 111 Final Exam Review: Comprehensive Study Notes for College Algebra

Ch. R - Review of Basic Concepts

Types of Numbers

Understanding the classification of numbers is foundational in algebra. Numbers are grouped into several categories based on their properties.

• Natural Numbers: Counting numbers starting from 1 (1, 2, 3, ...).

• Whole Numbers: Natural numbers including zero (0, 1, 2, 3, ...).

• Integers: Whole numbers and their negatives (..., -2, -1, 0, 1, 2, ...).

• Rational Numbers: Numbers that can be expressed as a fraction of two integers (e.g., 1/2, -3/4, 5).

• Irrational Numbers: Numbers that cannot be written as a simple fraction (e.g., \( \sqrt{2} \), \( \pi \)).

• Real Numbers: All rational and irrational numbers.

• Complex Numbers: Numbers in the form \( a + bi \), where \( i = \sqrt{-1} \).

Example: \( 2 \) is a natural, whole, integer, rational, and real number. \( \sqrt{2} \) is irrational and real. \( 3 + 2i \) is complex.

Properties of Equality

These properties are used to solve equations and maintain equality:

• Reflexive Property: \( a = a \)

• Symmetric Property: If \( a = b \), then \( b = a \)

• Transitive Property: If \( a = b \) and \( b = c \), then \( a = c \)

• Addition/Subtraction Property: If \( a = b \), then \( a + c = b + c \)

• Multiplication/Division Property: If \( a = b \), then \( ac = bc \) (and \( c \neq 0 \))

Exponent Rules

Exponent rules are essential for simplifying expressions:

• Product Rule: \( a^m \cdot a^n = a^{m+n} \)

• Quotient Rule: \( \frac{a^m}{a^n} = a^{m-n} \)

• Power Rule: \( (a^m)^n = a^{mn} \)

• Zero Exponent: \( a^0 = 1 \) (\( a \neq 0 \))

• Negative Exponent: \( a^{-n} = \frac{1}{a^n} \)

Example: \( 2^3 \cdot 2^4 = 2^{3+4} = 2^7 = 128 \)

Ch. 1 - Equations and Inequalities

Linear, Absolute Value, and Quadratic Equations and Inequalities

Solving and graphing equations and inequalities is a core skill in algebra.

• Linear Equations: Equations of the form \( ax + b = 0 \).

• Absolute Value Equations: Equations involving \( |x| \). Solve by considering both \( x \) and \( -x \).

• Quadratic Equations: Equations of the form \( ax^2 + bx + c = 0 \). Solutions can be found by factoring, completing the square, or using the quadratic formula:

$x = \frac{ -b \pm \sqrt{b^2 - 4ac} }{2a}$

• Inequalities: Similar to equations, but solutions are ranges of values. Remember to reverse the inequality sign when multiplying or dividing by a negative number.

Example: Solve \( 2x + 3 = 7 \): \( x = 2 \).

Forms of Linear and Quadratic Equations

• Slope-Intercept Form (linear): \( y = mx + b \)

• Standard Form (linear): \( Ax + By = C \)

• Vertex Form (quadratic): \( y = a(x-h)^2 + k \)

• Standard Form (quadratic): \( y = ax^2 + bx + c \)

Finding the Vertex: For \( y = ax^2 + bx + c \), the vertex is at \( x = -\frac{b}{2a} \).

Complex Numbers

Complex numbers are numbers of the form \( a + bi \), where \( i = \sqrt{-1} \).

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

• Multiplication: Use distributive property and \( i^2 = -1 \).

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

Example: \( (2 + 3i) + (1 - 2i) = 3 + i \)

Ch. 2 - Graphs and Functions

Key Characteristics of Functions and Graphs

Understanding the behavior of functions is crucial for graph analysis.

• Domain: Set of all possible input values (x-values).

• Range: Set of all possible output values (y-values).

• Maximum/Minimum: Highest/lowest point on the graph.

• Intervals of Increase/Decrease/Constant: Where the function rises, falls, or stays the same.

• Zeros/Roots/Solutions: Values of \( x \) where \( f(x) = 0 \).

• One-to-One: Each output is paired with exactly one input.

• Symmetry: Even (y-axis), odd (origin), or neither.

• x-intercept/y-intercept: Where the graph crosses the axes.

Example: The function \( f(x) = x^2 \) has domain all real numbers, range \( [0, \infty) \), minimum at (0,0), and is symmetric about the y-axis.

Function Operations and Composition

• Addition: \( (f+g)(x) = f(x) + g(x) \)

• Subtraction: \( (f-g)(x) = f(x) - g(x) \)

• Multiplication: \( (fg)(x) = f(x) \cdot g(x) \)

• Division: \( \left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} \), \( g(x) \neq 0 \)

• Composition: \( (f \circ g)(x) = f(g(x)) \)

Example: If \( f(x) = x+1 \) and \( g(x) = x^2 \), then \( (f \circ g)(2) = f(g(2)) = f(4) = 5 \).

Ch. 3 - Polynomials and Rational Functions

Polynomials

Polynomials are expressions of the form \( a_nx^n + a_{n-1}x^{n-1} + ... + a_0 \).

• Degree: Highest power of \( x \).

• Leading Coefficient: Coefficient of the highest power.

• End Behavior: Determined by degree and leading coefficient.

Example: \( 2x^3 - x^2 + 5 \) is a cubic polynomial.

Rational Functions

Rational functions are ratios of polynomials.

• Domain: All real numbers except where the denominator is zero.

• Asymptotes: Vertical (denominator zero), horizontal (degree comparison), or oblique.

Example: \( f(x) = \frac{1}{x-2} \) has a vertical asymptote at \( x=2 \).

Ch. 4 - Inverse, Exponential, and Logarithmic Functions

Inverse Functions

An inverse function reverses the effect of the original function.

• Notation: \( f^{-1}(x) \)

• Finding the Inverse: Swap \( x \) and \( y \), then solve for \( y \).

Example: For \( f(x) = 2x + 3 \), the inverse is \( f^{-1}(x) = \frac{x-3}{2} \).

Exponential and Logarithmic Equations

• Exponential Equations: Equations where the variable is in the exponent, e.g., \( a^x = b \).

• Logarithmic Equations: Equations involving logarithms, e.g., \( \log_a(x) = b \).

• Relationship: \( y = a^x \) is equivalent to \( x = \log_a(y) \).

Solving Exponential Equations: Take the logarithm of both sides if necessary.

Solving Logarithmic Equations: Rewrite in exponential form.

Example: Solve \( 2^x = 8 \): \( x = 3 \).

Ch. 5 - Systems and Matrices

Systems of Equations

Systems of equations involve finding values that satisfy multiple equations simultaneously.

• Methods: Substitution, elimination, and using matrices.

• Types of Solutions: One solution (consistent, independent), infinitely many (consistent, dependent), or none (inconsistent).

Example: Solve \( x + y = 5 \), \( x - y = 1 \): \( x = 3, y = 2 \).

Matrices

Matrices are rectangular arrays of numbers used to organize data and solve systems.

• Matrix Operations: Addition, subtraction, scalar multiplication, and matrix multiplication.

• Inverse Matrix: Used to solve systems: \( A\mathbf{x} = \mathbf{b} \Rightarrow \mathbf{x} = A^{-1}\mathbf{b} \).

• Determinant: Used to determine if a matrix is invertible.

Example: For \( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \), the determinant is \( 1 \times 4 - 2 \times 3 = -2 \).

Additional Topics

Distance and Midpoint

• Distance on a Number Line: \( |x_2 - x_1| \)

• Distance in the Plane: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Midpoint Formula: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

Example: The distance between (1,2) and (4,6) is $\sqrt{(4-1)^2 + (6-2)^2} = 5$.

Using Data Charts/Tables and Models

Data tables and mathematical models can be used to predict values of functions at given points. This is common in real-world applications such as predicting outcomes in sports or science.

• Identify the model (linear, quadratic, exponential, etc.).

• Use the model equation to substitute the given value and solve for the unknown.

Example: If a baseball's height is modeled by \( h(t) = -16t^2 + 64t + 5 \), find the height at \( t = 2 \): \( h(2) = -16(4) + 64(2) + 5 = -64 + 128 + 5 = 69 \).