Unit 3: Functions

3.1 Relations and Functions

Relation – any set of points

Function – a relation where every coordinate has a unique x-value

(A function will pass a vertical line test)

Domain – the set of x-values belonging to the relation

Range – the set of y-values belonging to the relation

Function Notation

A function is like a machine or a program. It takes the input (x) and converts it into something else (y). The function is the set of points (x, y) that are generated by taking all the x values and converting them into their respective y values. The function values are always the y values. This is why you often see a function defined as . A more sophisticated way of discussing a function is to write the coordinates as .

There are several ways to represent a function:

means f such that x is mapped onto 2x + 3 or the simpler .

Whatever is inside the brackets of the f replaces all of the same thing on the other side of the equals sign.

For example, if your function is defined as:

then

Another way to represent a function is with a mapping. A mapping uses ovals to describe how the domain and range interact.

For , where

Types of Functions we'll be covering:

-Linear

-Quadratic

-Exponential

-Trigonometric or etc.

-Higher Polynomials etc.

-Hyperbola

Which ones have you seen before?

Transformations on Functions

is a general way to write a function so that it's transformations are explicitly seen.

c – the vertical translation. c>0 means the function gets moved up.

b – the horizontal translation, notice that b is written with a negative sign in the brackets. b>0 means the function gets moved left.

a – the vertical stretch/reflection. means the graph gets stretched, a<0 means the graph gets reflected.

As many questions as you want from 9A-L