Transcript Inverse Relations and Inverse Functions
1.4c Inverse Relations and Inverse Functions
Homework: p. 129 39-61 odd
Definition: Inverse Relation
The ordered pair (a, b) is in a relation if and only if the ordered pair (b, a) is in the inverse relation.
Consider the “Do Now”:
y
x
2 Which of these relations are functions???
x
y
2 These relations are inverses of each other!
(the x- and y-values are simply switched!)
The HLT!!!
The Horizontal Line Test
The inverse of a relation is a function if and only if each horizontal line intersects the graph of the original function in at most one point.
Fails the HLT miserably!!!
So, its inverse is not a function…
Practice Problems
Is the graph of each relation a function? Does the relation have an inverse that is a function?
Not a function A function Has an inverse that is a function Has an inverse that is a function
Practice Problems
Is the graph of each relation a function? Does the relation have an inverse that is a function?
A function Not a function Has an inverse that is not a function Has an inverse that is not a function
More Definitions
A relation that passes both the VLT and HLT is called
one-to-one
.
(since every
x
is paired with a unique
y
with a unique
x
…) and every
y
is paired If
f
inverse function of f
, denoted
f
and range D defined by
b f
1 , is the function with domain R
a
if and only if
Finding an Inverse Algebraically
1. Determine that there is an inverse function by checking that the original function is one-to-one. Note any restrictions on the domain of the function.
2. Switch
x
and
y
in the formula of the original function.
3. Solve for
y
to obtain the inverse function. State any restrictions of the domain of the inverse.
Finding an Inverse Algebraically
Find the inverse of the given function algebraically:
x x
1 Check the graph is the function one-to-one?
x
y y
1
y y
x
x
1
x xy xy y x y
1
x x
Finding an Inverse Graphically The Inverse Reflection Principle
The points (
a
,
b
) and (
b
,
a
) in the coordinate plane are symmetric with respect to the line
y
=
x
. The points (
a
,
b
) and (
b
,
a
) are
reflections
of each other across the line
y
=
x
.
Finding an Inverse Graphically
The graph of a function is shown. Is the function one-to-one?
Sketch a graph of the inverse of the function.
y
=
x
Yes!!!
f
1
And one more new tool: The Inverse Composition Rule
A function
f
is one-to-one with inverse function
g
if and only if
f
(
g
(
x
)) =
x
for every
x
in the domain of
g
, and
g
(
f
(
x
)) =
x
for every
x
in the domain of
f
We can use this rule to algebraically verify that two functions are inverses… observe…
More Practice
Show algebraically that the given functions are inverses.
x
3 1 3
x
1 3 3
x
x
3 1 3 1 1
x
3
x
3
x
1
x
More Practice
Show that the given function has an inverse and find a rule for that inverse. State any restrictions of the domains of the function and its inverse.
x
3 Check the graph Is
f
one-to-one?
x y
x
2
y
x
3
y
3 3
x
2 3 where where where where
x
3,
y
0
y
3,
x
0
y
3,
x
0
y
3,
x
0
Let’s graph the inverse together with the original function…
Whiteboard Problems…
Find a formula for . Give the domain of , including any restrictions “inherited” from f.
f
1 2
x
5
f
1 1 2
x
5 2
D
: ( 3
x
2
f
1
x
3 2
D
: (
Whiteboard problems…
Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x.
x
3 4
x
4 3