Relation
a rule that associates the elements of one set with those of another set; can also be thought of as all of the ordered pairs which satisfy the rule; associates pairs of quantities by some rule of correspondence
Function
from a set A to a set B, a relation that assigns to each element x in the set A exactly one element y in the set B; a relationship between two variables such that each value of the independent variable there corresponds exactly one value of the dependent variable
Domain
set of inputs
Range
set of outputs
Domain
x-values
Range
y-values
Constant functions
regardless of the value of x, the y value always remains the same
Independent variable
x is what kind of variable?
Dependent variable
y is what kind of variable?
Domain
set of all values taken on by the independent variable
Range
set of all values taken on by the dependent variable
Function notation
special notation where one variable is written in terms of another, such as f(x)
y=f(x)
the y-value as written in function notation
Piecewise function
the function is defined by two or more equations over a specified domain
Implied domain
the set of all real numbers for which the expression is defined; avoid negative roots and values where that make the denominator zero
Difference quotient
f(x+h)-f(x)/h, h cannot equal zero
f
the name of a function
Graph of a function
the collection of ordered pairs (x,f(x)) such that x is in the domain of f
Vertical line test
easy way to determine if a relation is a function; at most one y-value corresponds to a given x-value; a set of points in a coordinate plane is the graph of y as a function of x if and only if no vertical line intersects the graph at more than one point
Greatest integer function
|| x || denotes this type of function, a type of step function; infinite number of breaks or steps- one at each integer value in its domain; jumps vertically one unit at a time and is constant between each pair of consecutive integers
All integers
What is the range of the greatest integer function?
Y-axis
Symmetry exists with this when (x,y) is on the graph as well as (-x,y)
X-axis
Symmetry exists with this when (x,y) is on the graph as well as (x,-y)
Origin
Symmetry exists with this when (x,y) is on the graph as well as (-x,-y)
Even function
a function whose graph is symmetric with respect to the y-axis
Odd function
a function whose graph is symmetric with respect to the origin
Not a function
If a graph has symmetry over the x-axis, it is said to be
V
What does the graph of an absolute value function look like?
Even
A function f is___ if, for each x in the domain of f, f(-x)=f(x).
Odd
A function f is___if, for each x in the domain f, f(-x)= -f(x).
Cubic function
f(x)=x^3
Quadratic function
f(x)=x^2
Square root function
f(x)=square root of x
Absolute value function
f(x)=lxl
Identity function
f(x)=x
Shifts and reflections
What are the two types of transformations that take place on a graph?
X-axis
Where does the reflection of h(x)= -f(x) take place?
Y-axis
Where does the reflection of h(x)=f(-x) take place?
Rigid transformations
What kind of transformations are horizontal shifts, vertical shifts, and reflections?
Rigid transformations
the basic shape of the graph remains unchanged in these types of transformations; only change position
Nonrigid transformations
the transformations that cause a distortion, a change in the shape of the original graph
Nonrigid transformations
vertical stretch and vertical shrink are examples of this type of transformation
Vertical stretch
f(x)=3lxl vertical stretch or vertical shrink?
Vertical shrink
f(x)=1/3lxl vertical stretch or vertical shrink?
Composition
This type of function is denoted by f open circle g
Inverse function
f^-1
Identity function
What do you obtain when you combine inverse functions into a composition function?
y=x
What is the equation of the line of reflection on the graph of inverse functions?
No
Does a function have to have an inverse?
One-to-one
To have an inverse, a function must be___, which means that no two elements in the domain of f correspond to the same element in the range of f.
One-to-one
If f is increasing or decreasing on its entire domain, it is what type of function?