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