1. Inverse Relations and Functions.
What you’ll learn
To find the inverse of a relation or function
Vocabulary
Inverse relation, inverse function,
one to one function

2. Did you remember? What is a relation?
A relation is simply a set of ordered pairs.
What is function?
A function is a set of ordered pairs in which each x-element has only ONE y-element associated with it.
What are the domain and the range in a function?
The first elements in the ordered pairs (the x-values), form the domain.  The second elements in the ordered pairs (the y-values), form the range.  Only the elements "used" by the relation constitute the range.
If a relation pairs element a of its domain to element b
of its range, the inverse relation pairs b with a. So if (a,b)
is an ordered pair of a relation, then (b,a) is an ordered
pair of its inverse. If both a relation and its inverse happen
to be functions, they are inverse functions.

3. What is the inverse of relation s?
Take a note
If a relation pairs element a of its domain to element b
of its range, the inverse relation pairs b with a. So if (a,b) is an ordered pair of a relation, then (b,a) is an ordered pair of its inverse. If both a relation and its inverse happen to be functions, they are inverse functions.
The inverse of a function may or may not be a function
Problem 1: Finding the Inverse of a Relation
What is the inverse of relation s?
Inverse of
relations s
Relation s x y -1 2 3 4 x y -1 2 3 4
Switch the x and the y
to get the inverse

4. B. What are the graphs of s and its inverse?
Relation s
Reversing the
Ordered Pairs
Inverse of s
The graph of a relation and its inverse are the
reflections of each other line y=x. If you describe
a relation or function by an equation in x and y, you
can switch x and y to get an equation for the inverse.

5. Problem 2: Finding an Equation for the Inverse.
What is the inverse of the relation described by
Switch x and y
Add 1 to each side
Find the square root of
each side to solve for y.
Your turn
Answer:

6. Problem 3:Graphing a Relation and its Inverse.
What are the graphs of and its inverse,
This graph is a translation of
down one unit.
It is a parabola that opens up.
The graph of the inverse is the
reflection of the parabola in the
line y=x. the slope of the
function is the reciprocal
of its inverse
The inverse of a function f is denoted by as the
inverse of. Sometimes this relation may not even
be a function.

7. Your turn
What is the graphs of y=2x+8 and its inverse?
Answer:

8. Problem 4: Finding an Inverse Function
Consider the function
What are the domain and range of f(x).
The radicand can not be negative, so the numbers
make up the domain. The principal square root
is nonnegative, do the numbers make the range.
B. What is ,the inverse of f?
Rewrite the eq. using y
Switch x and y.
Square both sides
Solve for y

9. C. What are the domain and the range of ?
The domain of is the range of f, numbers
Since Therefore, the numbers
make up the range of Note that the range of
is the same as the domain of f.
D.Is a function? Explain
For each x in the domain , there is
only one value of y in the range. So
is a function. Or you can use the vertical line test
as an alternate method.

10. Note:
For any function f, each x-value in the domain
corresponds to exactly one y-value in the range. For
one to one function, it is also true that each y-value
in the range corresponds to exactly one x-value in the
domain.
Composition of the inverse functions

11. Problem 5: Composing Inverse Functions.
For what is each of the following?
Answer

12. Your turn Answer 1 is not in the domain of f. Therefore
does not exist.
Your turn
Answer

13. Class work odd Homework even
TB pg exercises 8-47