College Algebra Study Guide: Equations, Inequalities, Functions, and Graphs

Equations & Inequalities

Solving Linear Inequalities

Linear inequalities are mathematical statements involving a variable where the two sides are related by inequality symbols such as <, >, ≤, or ≥. The solution to an inequality is a set of values that make the statement true.

• Key Point 1: To solve a linear inequality, isolate the variable using algebraic operations similar to those used for equations.

• Key Point 2: When multiplying or dividing both sides of an inequality by a negative number, reverse the direction of the inequality symbol.

• Example: Solve $x > 12$. The solution set is $(12, \infty)$.

Graphing Solution Sets

• Graph solutions on a number line using open or closed circles to indicate whether endpoints are included (closed for ≤ or ≥, open for < or >).

• Shade the region representing the solution set.

Compound Inequalities

Compound inequalities involve two inequalities joined by 'and' or 'or'. The solution is the intersection (for 'and') or union (for 'or') of the individual solution sets.

• Key Point: Solve each part separately, then combine the results.

• Example: Solve $1 < x \\leq 5$. The solution set is $(1, 5]$.

Graphs of Equations

Graphing Linear Equations

Linear equations can be graphed on the coordinate plane. The graph of a linear equation is a straight line.

• Key Point: The slope-intercept form of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Example: Graph $y = 2x + 1$ by plotting the y-intercept (0,1) and using the slope to find another point.

Graphing Solution Sets of Inequalities

Solution sets of inequalities can be represented on a number line or coordinate plane. For compound inequalities, shade the region that satisfies all parts.

• Key Point: Use interval notation to express solution sets, e.g., $(-\infty, 3]$.

Functions

Definition of a Function

A function is a relation in which each input (domain value) corresponds to exactly one output (range value).

• Key Point: The vertical line test can be used to determine if a graph represents a function: if any vertical line crosses the graph more than once, it is not a function.

• Example: The graph of a circle fails the vertical line test and is not a function.

Domain and Range

The domain of a function is the set of all possible input values, and the range is the set of all possible output values.

• Key Point: Use interval notation to express domain and range, e.g., $[0, 5)$.

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

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals of the domain.

• Key Point: Evaluate piecewise functions by determining which interval the input belongs to.

• Example: $f(x) = \begin{cases} x+2 & x < 0 \\ 2x-1 & x \geq 0 \end{cases}$

Function Notation and Evaluation

Function notation $f(x)$ indicates the output of function $f$ for input $x$. To evaluate, substitute the value of $x$ into the function's formula.

• Example: If $f(x) = 2x + 3$, then $f(4) = 2(4) + 3 = 11$.

Difference Quotient

The difference quotient is a formula used to compute the average rate of change of a function over an interval.

• Formula: $\frac{f(x+h) - f(x)}{h}$

• Example: For $f(x) = x^2$, $\frac{f(x+h) - f(x)}{h} = \frac{(x+h)^2 - x^2}{h} = \frac{2xh + h^2}{h} = 2x + h$

Properties of Functions

Increasing, Decreasing, and Constant Intervals

A function is increasing on an interval if its output values rise as the input increases, decreasing if they fall, and constant if they remain unchanged.

• Key Point: Use the graph to identify intervals of increase, decrease, or constancy.

• Example: If $f(x)$ rises from $x=1$ to $x=3$, it is increasing on $(1,3)$.

Relative Maximum and Minimum

A relative maximum is a point where the function reaches a peak locally, and a relative minimum is a local trough.

• Key Point: Use the graph or calculus to find these points.

• Example: For $f(x) = x^2 - 4x + 3$, the vertex at $x=2$ is a minimum.

Symmetry: Even and Odd Functions

A function is even if $f(-x) = f(x)$ for all $x$ in the domain (symmetric about the y-axis), and odd if $f(-x) = -f(x)$ (symmetric about the origin).

• Key Point: Test symmetry algebraically or by inspecting the graph.

• Example: $f(x) = x^2$ is even; $f(x) = x^3$ is odd.

Linear Equations and Graphs

Slope and Intercept

The slope of a line measures its steepness, and the y-intercept is where the line crosses the y-axis.

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

• Example: For points (1,2) and (3,6), $m = \frac{6-2}{3-1} = 2$

Equation Forms

• Slope-intercept form: $y = mx + b$

• Point-slope form: $y - y_1 = m(x - x_1)$

• General form: $Ax + By = C$

Graphing Lines

To graph a line, plot the y-intercept and use the slope to find another point.

• Key Point: Vertical lines have equations $x = a$; horizontal lines have equations $y = b$.

Tables: Interval Notation and Graphical Representation

Interval Notation Table

Interval notation is used to express solution sets and domains/ranges of functions.

Symbol	Interval Notation	Graphical Representation
<	$(a, b)$	Open circles at $a$ and $b$
≤	$[a, b]$	Closed circles at $a$ and $b$
>	$(b, \infty)$	Open circle at $b$, arrow to the right
≥	$[b, \infty)$	Closed circle at $b$, arrow to the right

Additional info:

• Some questions involve interpreting graphs to determine function properties, such as intervals of increase/decrease, maxima/minima, and symmetry.

• Piecewise functions and difference quotients are included, which are foundational for understanding calculus concepts.

• Questions on slope, intercept, and graphing lines are essential for understanding linear relationships in algebra.