Precalculus Study Notes: Relations, Functions, and Linear Equations

Introduction to Relations and Functions

Definitions and Set Notation

Understanding relations and functions is foundational in precalculus. The language of set notation and interval notation is used to describe domains and ranges of functions.

• Relation: A set of ordered pairs.

• Domain: The set of all possible input values (x-values).

• Range: The set of all possible output values (y-values).

• Function: A relation in which each element in the domain corresponds to exactly one element in the range.

Intervals are often described using set-builder notation, interval notation, and visualized on a number line.

Example: For F = {(-5, 6), (3, 7), (-5, 10)}, the domain is {-5, 3} and the range is {6, 7, 10}. F is not a function because -5 is paired with two different y-values.

Vertical Line Test

The vertical line test is a graphical method to determine if a relation is a function. If any vertical line intersects the graph more than once, the relation is not a function.

Key Point: Only graphs that pass the vertical line test represent functions.

Linear Functions and Equations

Forms of Linear Equations

Linear functions are described by equations of the form $f(x) = ax + b$, where $a$ and $b$ are real numbers. There are several standard forms:

• Standard Form: $Ax + By = C$

• Slope-Intercept Form: $y = mx + b$

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

To find intercepts:

• x-intercept: Set $y = 0$ and solve for $x$.

• y-intercept: Set $x = 0$ and solve for $y$.

Zero of a Function: A value $c$ such that $f(c) = 0$; corresponds to the x-intercept.

Slope of a Line

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

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

Example: Given a table of values, calculate the slope and write the equation in point-slope form.

Average Rate of Change

The average rate of change of a function $f$ on $[a, b]$ is:

$\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}$

This measures how much $y$ changes per unit change in $x$.

Piecewise-Defined Functions

Definition and Graphical Representation

A piecewise function is defined by different expressions over different intervals of its domain. Each piece applies to a specific part of the domain.

• Example: $f(x) = \begin{cases} 4 - x & x < 2 \\ 1 + 2x & x \geq 2 \end{cases}$

To evaluate a piecewise function, determine which interval the input belongs to and use the corresponding formula.

Graphs of Basic Functions

Parent Functions

Several basic functions serve as building blocks for more complex functions. Their graphs and properties are essential knowledge.

• Constant Function: $f(x) = c$

• Identity Function: $f(x) = x$

• Squaring Function: $f(x) = x^2$

• Cubing Function: $f(x) = x^3$

• Square Root Function: $f(x) = \sqrt{x}$

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

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

Key Points: Each parent function has a characteristic shape, domain, and range.

Even and Odd Functions

Definitions and Symmetry

Even Function: $f(-x) = f(x)$ for all $x$ in the domain. The graph is symmetric with respect to the y-axis. Odd Function: $f(-x) = -f(x)$ for all $x$ in the domain. The graph is symmetric with respect to the origin.

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

Transformations of Functions

Vertical and Horizontal Shifts

Transformations move or change the shape of a graph:

• Vertical Shift: $y = f(x) + c$ shifts up by $c$ units; $y = f(x) - c$ shifts down by $c$ units.

• Horizontal Shift: $y = f(x - c)$ shifts right by $c$ units; $y = f(x + c)$ shifts left by $c$ units.

Example: $y = |x - 3| + 1$ is the absolute value function shifted right 3 units and up 1 unit.

Vertical Stretching, Shrinking, and Reflection

Vertical Stretch: $y = c f(x)$, $c > 1$, stretches the graph vertically. Vertical Shrink: $0 < c < 1$, compresses the graph vertically. Reflection: $y = -f(x)$ reflects across the x-axis; $y = f(-x)$ reflects across the y-axis.

Operations and Composition of Functions

Operations

Given functions $f$ and $g$:

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

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

• Multiplication: $(fg)(x) = f(x)g(x)$

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

Composition

The composition $(g \circ f)(x) = g(f(x))$ means to apply $f$ first, then $g$ to the result.

Quadratic Functions and Their Graphs

Standard and Vertex Form

A quadratic function is $f(x) = ax^2 + bx + c$, $a \neq 0$. The vertex form is $f(x) = a(x - h)^2 + k$.

• Vertex: $(h, k)$, where $h = -\frac{b}{2a}$ and $k = f(h)$

• Axis of Symmetry: $x = h$

• Opens Up: $a > 0$; Opens Down: $a < 0$

Example: For $f(x) = 2x^2 - 8x + 9$, the vertex is at $x = 2$, $y = 2(2)^2 - 8(2) + 9 = 8 - 16 + 9 = 1$.

End Behavior of Polynomial Functions

Leading Term and Degree

The end behavior of a polynomial function is determined by its leading term $a_n x^n$:

• If $n$ is even and $a_n > 0$, both ends rise.

• If $n$ is even and $a_n < 0$, both ends fall.

• If $n$ is odd and $a_n > 0$, left end falls, right end rises.

• If $n$ is odd and $a_n < 0$, left end rises, right end falls.

Example: $f(x) = x^3$ has left end down, right end up.

Summary Table: Interval Notation

Types of Intervals

Intervals are used to describe sets of numbers, especially domains and ranges. The table below summarizes the main types:

Key: Parentheses ( ) indicate endpoints are not included; brackets [ ] indicate endpoints are included.