Precalculus Review: Functions, Graphing, Factoring, and Linear Equations

0. Fundamental Concepts of Algebra

0.1 Functions and Notation

Understanding functions and their notation is foundational in algebra and precalculus. A function is a rule that assigns to every element in a domain exactly one element in a range. Functions can be represented in various ways, including algebraic formulas, tables, graphs, and verbal descriptions.

• Function Notation: If a function is named f, and its input is x, we write the output as f(x). For example, $f(x) = x^2$ means that the function squares its input.

• Domain and Range: The domain is the set of all possible inputs (x-values), and the range is the set of all possible outputs (f(x)-values).

• Evaluating Functions: To evaluate a function at a specific value, substitute the value for the variable in the function's formula. For example, if $f(x) = 2x + 1$, then $f(3) = 2(3) + 1 = 7$.

Example: If $g(x) = x^2 - 4x + 2$, then $g(2) = (2)^2 - 4(2) + 2 = 4 - 8 + 2 = -2$.

3. Functions & Graphs

0.2 Function Evaluation Exercises

Several basic types of functions are commonly used in precalculus, including linear, quadratic, square root, and absolute value functions. Understanding their forms and how to evaluate them is essential.

• Linear Function: $f(x) = mx + b$ (straight line; constant rate of change)

• Quadratic Function: $f(x) = ax^2 + bx + c$ (parabola; variable rate of change)

• Square Root Function: $f(x) = \sqrt{x}$ (outputs the non-negative root of the input)

• Absolute Value Function: $f(x) = |x|$ (outputs the non-negative value of the input)

Example: For $h(x) = |x - 2|$, $h(5) = |5 - 2| = 3$.

2. Graphs

0.3 Graphing

Functions can be represented graphically, which helps visualize their behavior. The graph of a function is the set of all points $(x, f(x))$ in the coordinate plane.

• Graphing by Table of Values: Choose several x-values, compute the corresponding y-values, plot the points $(x, y)$, and connect them smoothly.

• Reading Graphs: Always read graphs from left to right, as you would read a book.

Example: To graph $f(x) = x^2 + 2$, create a table:

x	f(x)
-2	6
-1	3
0	2
1	3
2	6

Plot these points and connect them to form a parabola.

4. Polynomial Functions

0.4 Factoring

Factoring is a key algebraic skill, especially for solving equations and simplifying expressions. The most common type is factoring trinomials of the form $ax^2 + bx + c$.

• Factoring Quadratics: To factor $x^2 + bx + c$, find two numbers that multiply to $c$ and add to $b$.

• Factoring with Leading Coefficient: For $ax^2 + bx + c$, find two numbers that multiply to $a \times c$ and add to $b$, then split the middle term and factor by grouping.

Example: Factor $x^2 + 5x + 6$:

• Find two numbers that multiply to 6 and add to 5: 2 and 3.

• So, $x^2 + 5x + 6 = (x + 2)(x + 3)$

Example: Factor $2x^2 + 7x + 3$:

• Multiply $a \times c = 2 \times 3 = 6$

• Find two numbers that multiply to 6 and add to 7: 6 and 1.

• Rewrite: $2x^2 + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (x + 3)(2x + 1)$

1. Equations and Inequalities

0.5 Solving Linear Equations

Solving linear equations is a fundamental skill in algebra. A linear equation in one variable has the form $ax + b = c$.

• Isolate the Variable: To solve for $x$, first subtract or add terms to get all $x$ terms on one side and constants on the other.

• Divide by the Coefficient: Divide both sides by the coefficient of $x$ to solve for $x$.

Example: Solve $2x + 1 = 5$:

• Subtract 1 from both sides: $2x = 4$

• Divide by 2: $x = 2$

Example: Solve $3x - 2 = 4$:

• Add 2 to both sides: $3x = 6$

• Divide by 3: $x = 2$

Additional info: These notes provide a concise review of foundational algebra concepts essential for success in precalculus, including function notation, graphing, factoring, and solving linear equations. Mastery of these topics is critical for understanding more advanced topics in the course.