PLACEHOLDER Functions and Graphs

Learning Objectives

Define what constitutes a function
Identify domain and range of functions
Understand different types of functions
Interpret and create function graphs

What is a Function?

A function is a relation where each input has exactly one output. We write this as $f(x) = y$ , where:

$x$ is the input (independent variable)
$y$ is the output (dependent variable)
$f$ is the function name

Function Notation

If $f(x) = 2x + 3$ , then:

$f(1) = 2(1) + 3 = 5$
$f(-2) = 2(-2) + 3 = -1$
$f(a) = 2a + 3$

Domain and Range

Domain: The set of all possible input values (x-values)
Range: The set of all possible output values (y-values)

Example

For $f(x) = \sqrt{x - 2}$ :

Domain: $x \geq 2$ (since we can't take the square root of negative numbers)
Range: $y \geq 0$ (square root is always non-negative)

Types of Functions

Linear Functions

f(x) = mx + b

Slope: $m$
y-intercept: $b$
Graph: Straight line

Quadratic Functions

f(x) = ax^2 + bx + c

Vertex form: $f(x) = a(x - h)^2 + k$
Vertex: $(h, k)$
Graph: Parabola

Other Important Functions

Absolute Value: $f(x) = |x|$
Square Root: $f(x) = \sqrt{x}$
Reciprocal: $f(x) = \frac{1}{x}$

Transformations

Starting with a parent function $f(x)$ :

Transformation	New Function	Effect
Vertical shift up	$f(x) + k$	Move up $k$ units
Vertical shift down	$f(x) - k$	Move down $k$ units
Horizontal shift left	$f(x + h)$	Move left $h$ units
Horizontal shift right	$f(x - h)$	Move right $h$ units
Vertical stretch	$af(x)$	Stretch by factor $a$
Horizontal stretch	$f(\frac{x}{b})$	Stretch by factor $b$

Key Concepts to Remember

Vertical Line Test: A graph represents a function if every vertical line intersects it at most once
Function Composition: $(f \circ g)(x) = f(g(x))$
Inverse Functions: If $f$ and $g$ are inverses, then $f(g(x)) = x$ and $g(f(x)) = x$

Practice Problems

Try working through these during class:

Find the domain of $f(x) = \frac{1}{x^2 - 4}$
Graph $g(x) = |x - 2| + 1$
If $f(x) = x^2$ and $g(x) = x + 3$ , find $(f \circ g)(x)$