Functions and Graphs - Theory & Concepts

Functions and Graphs

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.

Function Notation and Tests

Functions are typically denoted as

f(x)

, where

x

is the independent variable (input) and

f(x)

is the dependent variable (output, often

y

Core Concepts

Domain: The set of all possible $x$ values for which the function is defined.
Range: The set of all possible resulting $y$ values.
Vertical Line Test: A graph represents a function if and only if no vertical line intersects the graph at more than one point.

Interactive Visualizer

Visualize 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

Types of Functions

Function Classifications

One-to-One (Injective): Each output $y$ comes from exactly one input $x$ . Passed via the Horizontal Line Test.
Inverse Functions: If a function $f$ is one-to-one, it has an inverse $f^{-1}$ such that $f(f^{-1}(x)) = x$ .
Piecewise Functions: Functions defined by different formulas for different parts of the domain.

Piecewise Example

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. It represents chaining processes together.

Composition Notation

Written as $(f \circ g)(x)$ or $f(g(x))$ .
Read as " $f$ composed with $g$ " or " $f$ of $g$ of $x$ ".
Order Matters: Generally, $f(g(x)) \neq g(f(x))$ . The function closest to $x$ (the inner function) is evaluated first. The output of $g(x)$ becomes the new input for $f(x)$ .
Domain Restriction: The domain of $f(g(x))$ includes only those $x$ 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

Symmetry Rules

Even Function: Symmetric about the y-axis. $f(-x) = f(x)$ .
Odd Function: Symmetric about the origin. $f(-x) = -f(x)$ .

Function Transformations

Base functions can be shifted, stretched, or reflected:

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

$a$ (Vertical Scale): $|a| \gt 1$ stretches vertically. $a \lt 0$ reflects over the x-axis.
$b$ (Horizontal Scale): $|b| \gt 1$ compresses horizontally by $1/b$ .
$h$ (Horizontal Shift): Shifts right if $h \gt 0$ .
$k$ (Vertical Shift): Shifts up if $k \gt 0$ .

Interactive Visualizer

Explore how changing the parameters

a

b

h

, and

k

transforms the graph of standard functions.

Function Transformations

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

g(x) = x^2

a (Vertical Stretch)1

b (Horizontal Stretch)1

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. They are fundamental in science and engineering modeling, such as Hooke's Law (direct variation) or Boyle's Law (inverse variation).

Variation Types

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

Key Takeaways

Domain Checks: Always look for division by zero and even roots of negative numbers.
Inverse Functions: To find the inverse, swap $x$ and $y$ , then solve for $y$ . This process flips the graph over the line $y=x$ .
Even/Odd: "Even" functions kill negative inputs ( $f(-x)=f(x)$ ). "Odd" functions spit the negative sign out ( $f(-x)=-f(x)$ ).
Piecewise Functions: Always check which interval your input falls into before calculating.
Transformations: Inner changes ( $f(x-h)$ ) shift horizontally; outer changes ( $f(x)+k$ ) shift vertically.

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