Precalculus Study Guide: Functions, Equations, and Graphs

Number Sets and Classification

Types of Numbers

Numbers can be classified into several sets, each with specific properties and examples. Understanding these sets is foundational for precalculus.

• Natural Numbers: Counting numbers starting from 1. Examples: 1, 2, 3, 4, ...

• Whole Numbers: Natural numbers plus zero. Examples: 0, 1, 2, 3, ...

• Integers: Whole numbers and their negatives. Examples: ..., -3, -2, -1, 0, 1, 2, 3, ...

• Rational Numbers: Numbers that can be written as a fraction $\frac{p}{q}$, where $q \neq 0$. Examples: $\frac{1}{2}$, $-3$, $0.4$, $3.5$

• Irrational Numbers: Numbers that cannot be written as a fraction. Examples: $\sqrt{2}$, $\pi$, $e$

• Real Numbers: All rational and irrational numbers.

Example: Classify the numbers $-1.5$, $-\sqrt{5}$, $-0.42$, $-0.4$, $1.4$, $\sqrt{5}$ into the above sets.

Graphing Points and Intervals

Plotting Points and Quadrants

The coordinate plane is divided into four quadrants. Each point $(x, y)$ is located based on its x and y values.

• Quadrant I: $x > 0$, $y > 0$

• Quadrant II: $x < 0$, $y > 0$

• Quadrant III: $x < 0$, $y < 0$

• Quadrant IV: $x > 0$, $y < 0$

Example: Plot the points (2, 4) and (0, -3) and state their quadrants.

Interval Notation

Intervals describe sets of numbers between two endpoints.

Inequality	Interval Notation
$a < x < b$	$(a, b)$
$a \leq x \leq b$	$[a, b]$
$x < a$	$(-\infty, a)$
$x > b$	$(b, \infty)$
$x$ is a real number	$(-\infty, \infty)$

Example: Write the interval for $x$ between 1 and 5, inclusive: $[1, 5]$.

Distance and Midpoint

Pythagorean Theorem

The Pythagorean Theorem relates the sides of a right triangle:

$a^2 + b^2 = c^2$

Distance Formula: The distance between $(x_1, y_1)$ and $(x_2, y_2)$ is:

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

Midpoint Formula: The midpoint between $(x_1, y_1)$ and $(x_2, y_2)$ is:

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

Linear Equations and Graphs

Slope and Slope-Intercept Form

The slope of a line through $(x_1, y_1)$ and $(x_2, y_2)$ is:

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

• Positive slope: Line rises left to right.

• Negative slope: Line falls left to right.

• Zero slope: Horizontal line.

• Undefined slope: Vertical line.

Slope-Intercept Form: $y = mx + b$

• $m$ = slope

• $b$ = y-intercept

Example: For $f(x) = 3x + 5$, find the slope, y-intercept, domain, range, and zero of $f$.

Parallel and Perpendicular Lines

• Parallel lines have the same slope.

• Perpendicular lines have slopes that are negative reciprocals: $m_1 \cdot m_2 = -1$

Example: Find the equation of a line passing through (1, -3) and parallel to $5x - y = 4$.

Linear Regression

Linear regression finds the best-fit line for a set of data points, minimizing the sum of squared vertical distances.

Year	2008	2009	2010	2011	2012	2013	2014	2015
Debt ($)	7788	7564	8123	8367	8562	8806	9212	9555

Example: Find the regression line and interpret the correlation coefficient.

Equations and Inequalities

Types of Equations

• Conditional equation: Has one or more solutions, but not infinitely many.

• Contradiction: No solution.

• Identity: True for all values in the domain.

Example: Solve $-4x - 3(2x - 2x_1) = 7 + 2x$ and classify the equation.

Solving Inequalities

To solve inequalities, isolate the variable and express the solution in interval notation.

Example: Solve $4(x + 7) > x + 2$.

Linear Applications

Cost, Revenue, and Break-Even Analysis

• Cost equation: $C = \text{fixed cost} + (\text{variable cost}) \times x$

• Revenue equation: $R = (\text{price per item}) \times x$

• Break-even point: Where $C = R$

Example: If fixed cost is $40,000$ and variable cost is $3$per T-shirt, selling price is $10$, find the break-even quantity.

Functions and Their Properties

Basic Functions

• Linear function: $f(x) = x$

• Square function: $f(x) = x^2$

• Cube function: $f(x) = x^3$

• Square root function: $f(x) = \sqrt{x}$

• Cube root function: $f(x) = \sqrt[3]{x}$

• Absolute value function: $f(x) = |x|$

Continuity, Domain, and Range

A function is continuous if its graph can be drawn without lifting the pencil. The domain is the set of all possible input values, and the range is the set of all possible output values.

Example: Determine the domain and range of $f(x) = \sqrt{x}$.

Increasing, Decreasing, and Constant Intervals

• Increasing: $f(x_2) > f(x_1)$ for $x_2 > x_1$

• Decreasing: $f(x_2) < f(x_1)$ for $x_2 > x_1$

• Constant: $f(x_2) = f(x_1)$ for $x_2 > x_1$

Symmetry, Even and Odd Functions

• Even function: $f(-x) = f(x)$ (symmetric about the y-axis)

• Odd function: $f(-x) = -f(x)$ (symmetric about the origin)

Example: Determine if $f(x) = -2x^4 - 8x^3$ is even, odd, or neither.

Transformations of Functions

Types of Transformations

Transformation	Effect
Vertical Shift Up	$f(x) + c$
Vertical Shift Down	$f(x) - c$
Horizontal Shift Right	$f(x - c)$
Horizontal Shift Left	$f(x + c)$
Vertical Stretch	$cf(x)$, $c > 1$
Vertical Compression	$cf(x)$, $0 < c < 1$
Reflection about x-axis	$-f(x)$
Reflection about y-axis	$f(-x)$

Example: The graph of $y = -\frac{1}{4}|x + 2| - 3$ is obtained from $y = |x|$ by shifting left 2 units, reflecting across the x-axis, vertically shrinking by a factor of $\frac{1}{4}$, and shifting down 3 units.

Piecewise Functions

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

Example: $f(x) = \begin{cases} 2x - 3 & x < -1 \\ -2x + 1 & -1 \leq x < 0 \\ x^2 & 0 \leq x \leq 2 \end{cases}$

Absolute Value Equations and Inequalities

Absolute Value Equations

For $|x| = c$, $x = c$ or $x = -c$.

Example: Solve $|x - 1| = 5$.

Absolute Value Inequalities

• If $|x| < c$, then $-c < x < c$

• If $|x| > c$, then $x < -c$ or $x > c$

Example: Solve $|x - 1| < 5$ and express the answer in interval notation.

Operations with Functions

Function Operations

• Sum: $(f + g)(x) = f(x) + g(x)$

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

• Product: $(fg)(x) = f(x)g(x)$

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

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

Example: If $f(x) = x^2 + 3x$ and $g(x) = 5 - x$, find $(f + g)(x)$, $(f \circ g)(x)$, and their domains.

Difference Quotient

The difference quotient is used to compute the average rate of change:

$\frac{f(x + h) - f(x)}{h}$

Example: For $f(x) = x^2 - 2x + 7$, find and simplify the difference quotient.

Additional Applications

• Modeling cost, revenue, and profit with linear functions

• Solving real-world problems using systems of equations

• Analyzing graphs for domain, range, and intervals of increase/decrease

Example: Suppose you have $50,000$ to invest in stocks and bonds. Stocks pay $6\%$ annually, bonds pay $2\%$. If the annual interest income is $1,892$, how much was invested in each?