Functions - Summary and Quick Reference

Functions: Summary and Quick Reference

Article Contents

Core Concepts
Types of Functions
Domain and Range Quick Reference
Graph Shapes
Key Formulas
How to Draw a Graph
All Articles in This Topic

1. Core Concepts {#core-concepts}

Concept	Definition	Article
Function	A rule assigning exactly one output to each input	Introduction
$f (x)$ notation	$f$ = name, $x$ = input, $f (x)$ = output	Introduction
Independent variable	The input ( $x$ ); we choose it freely	Variables
Dependent variable	The output ( $y = f (x)$ ); determined by the rule	Variables
Domain $D (f)$	Set of all allowed inputs	Domain
Range $R (f)$	Set of all possible outputs	Range
x-intercept	Where $f (x) = 0$ (graph crosses x-axis)	Graphs
y-intercept	The value $f (0)$ (graph crosses y-axis)	Graphs

2. Types of Functions {#types-of-functions}

Type	Formula	Graph	Key property
Direct proportion	$y = k x$	Line through origin	$\frac{y}{x} = k$ constant
Linear function	$y = k x + q$	Straight line	Slope $k$ , y-intercept $q$
Inverse proportion	$y = \frac{k}{x}$	Hyperbola	$x \cdot y = k$ constant
Constant function	$y = c$	Horizontal line	Output never changes
Quadratic	$y = a x^{2} + b x + c$	Parabola	Has a vertex (min or max)

3. Domain and Range Quick Reference {#domain-range}

Function	Domain $D (f)$	Range $R (f)$
$y = k x + q$ ( $k \neq = 0$ )	$R$	$R$
$y = c$	$R$	${c}$
$y = k x$ ( $k \neq = 0$ )	$R$	$R$
$y = \frac{k}{x}$	$R ∖ {0}$	$R ∖ {0}$
$y = x^{2}$	$R$	$[0, + \infty)$
$y = x$	$[0, + \infty)$	$[0, + \infty)$

Domain restrictions:

Denominator $\neq = 0$
Expression under $x \geq 0$

4. Graph Shapes {#graph-shapes}

5. Key Formulas {#key-formulas}

What	Formula
Slope from two points	$k = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$
Linear equation	$y = k x + q$
Y-intercept	$q = y - k x$ (use a known point)
X-intercept of linear	$x = - \frac{q}{k}$
Direct proportion constant	$k = \frac{y}{x}$
Inverse proportion constant	$k = x \cdot y$

6. How to Draw a Graph {#how-to-draw}

Create a table with at least 2 points (more for curves)
Set up the coordinate system with appropriate scale
Plot each point $(x, f (x))$
Connect with a straight line (linear) or smooth curve
Label the graph

7. All Articles in This Topic {#all-articles}

#	Article	Description
1	Introduction to Functions	What is a function, $f (x)$ notation, representations
2	Variables in Functions	Independent vs dependent variables
3	Functions and Tables	Tables as function representation
4	Functions and Graphs	Coordinate system, plotting, reading graphs
5	Domain of a Function	Allowed inputs, restrictions
6	Range of a Function	Possible outputs
7	Direct Proportion	$y = k x$ , line through origin
8	Inverse Proportion	$y = k / x$ , hyperbola
9	Linear Function	$y = k x + q$ , slope, y-intercept
10	Summary (this page)	Quick reference card