Key Concepts in College Algebra: Functions, Graphs, and Matrices

Functions and Their Representations

Definition of a Function

A function is a relation that assigns exactly one output value to each input value. Functions can be represented in various forms, including equations, tables, graphs, and mappings.

• Notation: $f(x)$ denotes a function named f with input variable x.

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

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

Example: The function $f(x) = x^2$ assigns to each real number x its square.

Graphical Representation of Functions

Functions can be visualized using graphs on the Cartesian coordinate plane. The graph of a function is the set of all points (x, f(x)).

• Key Features: Intercepts, increasing/decreasing intervals, maxima/minima, and asymptotes.

• Example: The graph of $f(x) = \sqrt{x}$ starts at (0,0) and increases slowly to the right.

Piecewise-Defined and Power Functions

Piecewise-Defined Functions

A piecewise-defined function is defined by different expressions for different intervals of the domain.

• Example: $f(x) = \begin{cases} x^2 & x < 0 \\ x & x \geq 0 \end{cases}$

Power Functions

A power function has the form $f(x) = ax^n$, where a and n are constants.

• Example: $f(x) = x^3$ is a cubic power function.

Graphs and Data Representation

Scatter Plots

A scatter plot is a graphical representation of data points on a coordinate plane, used to observe relationships between variables.

• Application: Scatter plots help identify trends, clusters, and outliers in data.

Graphing on the Coordinate Plane

The coordinate plane consists of a horizontal axis (x-axis) and a vertical axis (y-axis) intersecting at the origin (0,0).

• Quadrants: The plane is divided into four quadrants.

• Plotting Points: Each point is represented as (x, y).

Matrices and Systems of Equations

Matrix Notation and Operations

A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are used to organize data and solve systems of equations.

• Notation: A matrix is often denoted by a capital letter, e.g., $A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$

• Operations: Addition, subtraction, and multiplication are defined for matrices of appropriate sizes.

Solving Systems Using Matrices

Systems of linear equations can be represented and solved using matrices.

• Augmented Matrix: Represents a system of equations in matrix form.

• Row Operations: Used to simplify matrices and solve systems (Gaussian elimination).

Example: Solve the system: $\begin{cases} x + y = 2 \\ 2x - y = 1 \end{cases}$ Represented as an augmented matrix: $\begin{bmatrix} 1 & 1 & | & 2 \\ 2 & -1 & | & 1 \end{bmatrix}$

Summation Notation

Definition and Use

Summation notation (the Greek letter sigma $\sum$) is used to represent the sum of a sequence of terms.

• General Form: $\sum_{i=1}^{n} a_i$ means add all $a_i$ from $i=1$ to $i=n$.

• Example: $\sum_{k=1}^{4} k = 1 + 2 + 3 + 4 = 10$

Radicals and Roots

Square Roots and Radical Expressions

A radical expression involves roots, such as square roots or cube roots.

• Notation: $\sqrt{x}$ denotes the principal (non-negative) square root of x.

• Properties: $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$, for $a, b \geq 0$.

Example: $\sqrt{16} = 4$

Additional info: Some content and examples have been inferred and expanded for completeness due to the fragmentary nature of the original notes and images.