Functions and Graphs - Theory & Concepts

Functions and Graphs

Learning Objectives

Define functions, domain, and range.
Apply the Vertical Line Test to identify functions.
Understand one-to-one, inverse, piecewise, and composite functions.
Identify even and odd function symmetry.
Apply transformations to base functions.
Model proportional relationships using variation equations.

Function Definition

A function is a fundamental concept in mathematics describing a relationship between two sets: the input (domain) and the output (range). The defining characteristic of a function is that every input maps to exactly one output.

Domain

The set of all possible $x$ values (inputs) for which the function is defined.

Range

The set of all possible resulting $y$ values (outputs).

Vertical Line Test

A graph represents a function if and only if no vertical line intersects the graph at more than one point. This ensures that each input has exactly one output.

Core Concepts

Understanding the domain and range of a function is critical for graphing and evaluating inputs properly.

Interactive Simulation

Interactive visualizer to explore the relationship between inputs and outputs, and see how domain restrictions affect the graph.

Domain and Range Visualizer

Drag the blue point to explore inputs (Domain, x) and their resulting outputs (Range, y) for the function y = -0.5(x - 2)² + 4.

x = 2.0

y = 4.0

Input (Domain)

2.00

Output (Range)

4.00

One-to-One (Injective) Function

A function where each output $y$ comes from exactly one input $x$ . This type of function passes the Horizontal Line Test.

Inverse Functions

If a function $f$ is one-to-one, it has an inverse $f^{-1}$ such that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ .

Piecewise Functions

Functions that are defined by different mathematical formulas for different parts of their domain.

Function Classifications

Functions are classified based on their mapping properties and how they are defined across their domain.

Piecewise Function Example

An example of a piecewise function where the rule changes based on whether the input $x$ is negative or non-negative:

f(x) = \begin{cases} x + 1 & \text{if } x \lt 0 \\ x^2 & \text{if } x \ge 0 \end{cases}

Composite Functions

Function composition involves applying one function to the results of another, representing chaining processes together.

Notation: Written as $(f \circ g)(x)$ or $f(g(x))$ . It is read as " $f$ composed with $g$ " or " $f$ of $g$ of $x$ ".
Order of Operations: The inner function is evaluated first. The output of $g(x)$ becomes the new input for $f(x)$ . Generally, function composition is not commutative, meaning $f(g(x)) \neq g(f(x))$ .
Domain Restriction: The domain of $f(g(x))$ only includes $x$ values that are in the domain of $g(x)$ , AND for which the output $g(x)$ falls within the domain of $f(x)$ .

Symmetry: Even and Odd Functions

Functions can exhibit specific symmetrical properties:

Even Function: Symmetric about the y-axis. Mathematically, it satisfies the condition $f(-x) = f(x)$ .
Odd Function: Symmetric about the origin. Mathematically, it satisfies the condition $f(-x) = -f(x)$ .

Function Transformations

Base functions can be shifted, stretched, or reflected using a general transformation formula.

General Transformation Formula

Describes how a base function $f(x)$ is shifted, stretched, or reflected to create a new function $g(x)$ .

g(x) = a \cdot f(b(x - h)) + k

Variables

Symbol	Description	Unit
$g(x)$	The transformed function	-
$f(x)$	The original base function	-
$a$	Vertical stretch/compression and reflection over x-axis	-
$b$	Horizontal stretch/compression and reflection over y-axis	-
$h$	Horizontal shift (translation right or left)	-
$k$	Vertical shift (translation up or down)	-

Parameter Effects on Transformations

$a$ (Vertical Scale): If $|a| > 1$ , the graph stretches vertically. If $a \lt 0$ , it reflects over the x-axis.
$b$ (Horizontal Scale): If $|b| > 1$ , the graph compresses horizontally by a factor of $1/b$ .
$h$ (Horizontal Shift): Shifts the graph to the right if $h > 0$ .
$k$ (Vertical Shift): Shifts the graph up if $k > 0$ .

Interactive Simulation

Explore the effects of function transformations interactively. Adjust parameters $a$ , $b$ , $h$ , and $k$ to see vertical and horizontal scaling, reflections, and shifts on the parent function $y = f(x)$ .

Function Transformations

General Form: $g(x) = a \cdot f(b(x - h)) + k$

g(x) = x^2

Scaling & Reflection

a (Vertical Stretch)1

b (Horizontal Stretch)1

Translations

h (Horizontal Shift)0

k (Vertical Shift)0

Original:

f(x) = x^2

Transformed:

g(x)

Direct, Inverse, and Joint Variation

Variation equations describe how quantities relate to one another proportionally, which is fundamental in engineering modeling.

Direct Variation: Represented by the equation $y = kx$ . As $x$ increases, $y$ increases proportionally.
Inverse Variation: Represented by the equation $y = \frac{k}{x}$ . As $x$ increases, $y$ decreases.
Joint Variation: Represented by an equation like $z = kxy$ . A quantity varies directly with multiple other variables.

Domain Exclusions

When checking for valid domains, always ensure you avoid division by zero and taking the even root of a negative number, as these are the most common restrictions leading to undefined outputs or complex numbers.

Key Takeaways

A function requires that every input maps to exactly one output, verified by the Vertical Line Test.
Domain restrictions primarily arise from avoiding division by zero and avoiding even roots of negative numbers.
Inverse functions are found by swapping $x$ and $y$ , which geometrically reflects the graph across the line $y = x$ .
Function transformations shift and scale graphs: inner modifications affect horizontal position, while outer modifications affect vertical position.
Even functions are symmetric about the y-axis ( $f(-x) = f(x)$ ) and odd functions are symmetric about the origin ( $f(-x) = -f(x)$ ).

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