A Function is a relation where each x produces one, and
only one, y.
The Domain of a function are all of the x’s that are allowed
to be used.
The Range of a function are all the possible y’s that could
To determine the range of a function, say y = x^2, solve
For what y’s can you find x’s?
In this case, you can find x’s only for y’s bigger than or
equal to 0, so the range is all non-negative numbers in R .
We will learn an easier way to find the range in the next
What is the range of
Review: Graphing→Domain, Range
Simply scan across to find the values where the function
not defined at x. This is the domain.
Then scan up to find the values where the function is not
defined at y. This is the range.
So the domain is [-4,5] and the range is [-2,3) with
The average rate of change between x and x + h when h is
small will be very close to the actual rate of change at x
Most graphs are translations of easy graphs that we know.
Review: Effects of Transformations
•A > 0 “scales” the graph
•“scales” could be “stretching” (A > 1) or “squashing” (0 < A < 1)
•A < 0 “scales” the graph after “flipping” it over the x-axis.
•B > 0 “scales” the graph ↔
•B < 0 “scales” the graph after “flipping” it over the y-axis.
•h “shifts” the graph ↔
•k “shifts” the graph
Function iteration is just applying functions in sequence.
§3.6: Inverse Functions
An inverse function for f, usually written f−1, is the function
that asks the question, “What do I put into f to get y out?”
Finding inverse relations with Graphs
To find an inverse relationship, it should be clear that
switch the domain, x, with the range, y.
It is clear that this new graph is a “flip” across the
line y = x.
Inverse Graphs #2
Because switching the domain, x, with the range, y is a
over y = x we have:
Horizontal Line Test
If a function does not pass the “Horizontal line test”
can you say about its inverse?
A function f is “one-to-one” if each y has only one x
f(x) = y.
How does this relate to the Horizontal line test and the
inverse of f? Answer: A function has an inverse function if and only if it is
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