Key Concepts in College Algebra: Functions, Equations, and Factoring

Functions and Their Properties

Definition of a Function

A function is a relation that assigns exactly one output value to each input value. Functions are fundamental in algebra and are used to model relationships between quantities.

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

• Range: The set of all possible output values (usually y-values) that the function can produce.

Example: For the function $f(x) = x^2$, the domain is all real numbers, and the range is all real numbers greater than or equal to zero.

Linear Functions and Equations

Equations of Lines

Linear equations describe straight lines in the coordinate plane. The general form of a linear equation is:

$ax + by = c$

• Slope-Intercept Form: $y = mx + b$, where m is the slope and b is the y-intercept.

• Point-Slope Form: $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line.

Example: Find the equation of a line passing through points $(2, 3)$ and $(4, 7)$.

First, calculate the slope:

$m = \frac{7 - 3}{4 - 2} = 2$

Using point-slope form:

$y - 3 = 2(x - 2)$

Simplify to slope-intercept form:

$y = 2x - 1$

Quadratic Functions and Factoring

Factoring Polynomials

Factoring is the process of writing a polynomial as a product of its factors. This is a key skill for solving quadratic equations and simplifying expressions.

• Common Factoring Techniques:

  • Factoring out the greatest common factor (GCF)

  • Factoring trinomials: $ax^2 + bx + c$

  • Difference of squares: $a^2 - b^2 = (a - b)(a + b)$

Example: Factor $x^2 - 9$.

$x^2 - 9 = (x - 3)(x + 3)$

Solving Word Problems with Systems of Equations

Setting Up and Solving Systems

Many real-world problems can be modeled using systems of equations. To solve these problems:

1. Define variables to represent unknowns.

2. Write equations based on the relationships described in the problem.

3. Solve the system using substitution, elimination, or matrix methods.

Example: If two numbers add up to 10 and their difference is 2, find the numbers.

Let $x$ and $y$ be the numbers:

$\begin{cases} x + y = 10 \\ x - y = 2 \end{cases}$

Add the equations:

$2x = 12 \implies x = 6$

Substitute back:

$6 + y = 10 \implies y = 4$

Variable Expressions and Rational Functions

Working with Variable Fractions

Expressions with variables in the denominator are called rational expressions. Simplifying and factoring these expressions is important for solving equations and understanding function behavior.

• Example: Simplify $\frac{x^2 - 9}{x - 3}$.

Factor numerator:

$\frac{(x - 3)(x + 3)}{x - 3} = x + 3, \quad x \neq 3$

Additional info: The restriction $x \neq 3$ is necessary because the original denominator cannot be zero.