College Algebra: Core Concepts and Functions Study Guide

1. Equations and Inequalities

1.1 Solving Equations

Solving equations is a foundational skill in algebra, involving finding the value(s) of the variable(s) that make the equation true.

• Linear Equations: Equations of the form $ax + b = 0$.

• Quadratic Equations: Equations of the form $ax^2 + bx + c = 0$.

• Irrational Equations: Equations involving roots, such as $\sqrt{x} + 2 = 5$.

• Literal Equations: Equations involving multiple variables, solved for one variable in terms of others.

Example: Solve $2x + 3 = 7$.

Subtract 3 from both sides: $2x = 4$. Divide by 2: $x = 2$.

1.2 Solving Inequalities

Inequalities express a range of possible values for variables, rather than a single solution.

• Linear Inequalities: $ax + b > 0$ or $ax + b < 0$.

• Compound Inequalities: $a < x < b$.

• Absolute Value Inequalities: $|x| < a$ or $|x| > a$.

Example: Solve $x - 3 > 2$.

Add 3 to both sides: $x > 5$.

2. Relations and Circles

2.1 Distance and Midpoint Formulas

These formulas are used to find the distance between two points and the midpoint of a segment in the coordinate plane.

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

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

Example: Find the distance between $(1,2)$ and $(4,6)$.

$d = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

2.2 Equations of Circles

The equation of a circle in the coordinate plane can be written in standard or general form.

• Standard (Center-Radius) Form: $(x - h)^2 + (y - k)^2 = r^2$

• General Form: $x^2 + y^2 + Dx + Ey + F = 0$

• Completing the square can convert the general form to the center-radius form.

Example: Write $x^2 + y^2 - 4x + 6y - 12 = 0$ in center-radius form.

Complete the square for $x$ and $y$ to find the center and radius.

3. Functions

3.1 Key Definitions

Functions describe relationships between sets, assigning each input exactly one output.

• Function: A rule that assigns to each element in the domain exactly one element in the range.

• Dependent Variable: The output variable, often $y$.

• Independent Variable: The input variable, often $x$.

• Domain: The set of all possible input values.

• Range: The set of all possible output values.

Example: For $f(x) = x^2$, the domain is all real numbers, and the range is $[0, \infty)$.

3.2 Graphs of Functions

Graphs visually represent the relationship between variables in a function.

• Identify domain, range, intervals of increase/decrease, and constant intervals from the graph.

• Recognize features such as maxima, minima, and points of inflection.

Example: The graph of $f(x) = x^2$ is a parabola opening upward.

3.3 Linear Functions

Linear functions have graphs that are straight lines and can be represented in several forms.

• Slope-Intercept Form: $y = mx + b$

• Point-Slope Form: $y - y_1 = m(x - x_1)$

• Slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals.

Example: Find the equation of a line through $(2,3)$ with slope $4$.

$y - 3 = 4(x - 2)$

4. Basic Functions and Their Properties

4.1 Types of Functions

Several basic functions are commonly studied in algebra.

• Constant Function: $f(x) = c$

• Identity Function: $f(x) = x$

• Absolute Value Function: $f(x) = |x|$

• Piecewise Functions: Functions defined by different expressions over different intervals.

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

5. Graphs of Functions: Transformations

5.1 Types of Transformations

Transformations change the position or shape of a graph.

• Vertical Shifting: $f(x) + k$ shifts the graph up/down.

• Horizontal Shifting: $f(x - h)$ shifts the graph left/right.

• Vertical Stretching/Shrinking: $a f(x)$ stretches/shrinks vertically.

• Horizontal Stretching/Shrinking: $f(bx)$ stretches/shrinks horizontally.

• Reflections: $-f(x)$ reflects over the $x$-axis; $f(-x)$ reflects over the $y$-axis.

Example: The graph of $f(x) = (x - 2)^2 + 3$ is shifted right by 2 units and up by 3 units compared to $f(x) = x^2$.

6. Function Operations and Composition

6.1 Operations on Functions

Functions can be combined using addition, subtraction, multiplication, division, and composition.

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

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

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

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

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

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

6.2 Domain of Combined Functions

The domain of a function resulting from operations or composition is determined by the domains of the original functions and the operation performed.

• For $f(x) + g(x)$, the domain is the intersection of the domains of $f$ and $g$.

• For $\frac{f(x)}{g(x)}$, exclude values where $g(x) = 0$.

• For $f(g(x))$, the domain is all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$.

Example: If $f(x) = \sqrt{x}$ and $g(x) = x - 2$, then $f(g(x)) = \sqrt{x - 2}$ has domain $x \geq 2$.

7. Table: Types of Transformations

Transformation	Algebraic Form	Effect on Graph
Vertical Shift	$f(x) + k$	Up if $k > 0$, down if $k < 0$
Horizontal Shift	$f(x - h)$	Right if $h > 0$, left if $h < 0$
Vertical Stretch/Shrink	$a f(x)$	Stretch if $|a| > 1$, shrink if $0 < |a| < 1$
Horizontal Stretch/Shrink	$f(bx)$	Shrink if $|b| > 1$, stretch if $0 < |b| < 1$
Reflection over x-axis	$-f(x)$	Flips graph over x-axis
Reflection over y-axis	$f(-x)$	Flips graph over y-axis

8. Additional info:

• Some content inferred from standard College Algebra syllabi and textbook structure to ensure completeness.

• Examples and explanations expanded for clarity and self-contained study.