Functions

Evaluating in/out(puts) on a graph

- Points on a graph ar written as (x,y) or (input,output) - Write the given information as a point (x,y), use the graph to find the missing value

If tables/sets (mappings) represent a function

...no domain/input/x-values are repeated.

If a graph represent a function

...they will pass the vertical line test. If a vertical line crosses the graph more than once it is not a function because it means that domain (x-values) are repeated.

Input

Anything inside the ( ) represents the x value also known as the...

Increasing/Decreasing Intervals

Elements of the domain (x-values) where graph is rising/going down left to right ALWAYS USE PARENTHESIS when stating intervals, because endpoints are not included.

Function

Every input (x-value) has EXACTLY ONE output (y-value) No x values repeat Passes VLT

Composition of functions

Functions convert the value of an input variable into the value of an output variable. It stands to reason that this output could then be used as an input to a second function. This process is known as...

Absolute Max/Min

Highest and lowest y-values on a graph Written as numeric y-values

Maximum

Highest point (y-value) can be endpoint or turning point

Minimum

Lowest point (y-value) can be endpoint or turning point

Relative Max/Min (s)

Occur at turning points in the graph - never an endpoint Written as ordered pairs

Odd functions

Opposite INPUTS give OPPOSITE outputs f(-x) = -f(x) Graphs are symmetric to origin which is the same as a rotations around the origin of 180° (R o,180°)

Even functions

Opposite INPUTS give SAME outputs f(-x) = f(x) Graphs are symmetric to y-axis

Domain

Set of INPUTS (x-value) - independent variable

Range

Set of OUTPUTS (y-value) - dependent variable

Inverse of a function: [ f^-1(x) and (x,y) -> (y,x) ]

Swap x and y value then solve for y because they undo each other (new and original equations)

Output

The answer represents the y-value or f(x) also known as the...

HLT

To know if the inverse will be a function, apply the _______ to the original function. If it passes, the inverse will be a function

Interval notation

Uses ( ) for open circles (not included) or [ ] for closed circles (included) to show the set of all numbers between 2 endpoints

Set notation

Uses <, <_, _>, > to show the set of all numbers between 2 endpoints - < or > open circles - endpoints not included - <_ or _> closed circles - endpoints included

Multiple choice

Using you calculator to test function can only be used on _______ _______ questions

f(x)

We replace y with...

End behavior

What graph is doing (increasing or decreasing) as x -> -infinity and x -> infinity

Solving composition of functions

Work inside to outside 1.) Identify the input value and circle the function and input value - find the output for this function 2.) This output becomes the new input value for the outside function. Find the final output value. This will be your final answer.

Composition of function notation

f(g(x)) -> always work with the inner set of parentheses first (f°g)(x) -> open circle means following

One to One Function

no x or y values repeat passes VLT and HLT