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1.4c Inverse Relations and Inverse Functions
Homework: p. 129 39-61 odd
Definition: Inverse Relation
2y x2x y
The ordered pair (a, b) is in a relation if and only if theordered pair (b, a) is in the inverse relation.
Consider the “Do Now”:
These relations are inverses of each other!(the x- and y-values are simply switched!)
Which of these relationsare functions???
The HLT!!!
The Horizontal Line TestThe inverse of a relation is a function if and only if eachhorizontal line intersects the graph of the original functionin at most one point.
Fails the HLT miserably!!!
So, its inverse is not a function…
Practice ProblemsIs the graph of each relation a function? Does therelation have an inverse that is a function?
A function
Has an inverse thatis a function
Not a function
Has an inverse thatis a function
Practice ProblemsPractice ProblemsIs the graph of each relation a function? Does therelation have an inverse that is a function?
Not a function
Has an inverse thatis not a function
A function
Has an inverse thatis not a function
More DefinitionsMore Definitions
1f b a f a b
A relation that passes both the VLT and HLT is called one-to-one.
(since every x is paired with a unique y and every y is paired with a unique x…)
If f is a one-to-one function with domain D and range R, then theinverse function of f, denoted f , is the function with domain Rand range D defined by
–1
if and only if
Finding an Inverse AlgebraicallyFinding an Inverse Algebraically1. 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 AlgebraicallyFinding an Inverse Algebraically
1
xf x
x
1
yx
y
Find the inverse of the given function algebraically:
Check the graph isthe function one-to-one?
xy y x xy x y
( 1)y x x
1
xy
x
1
xy
x
1x y y
Finding an Inverse GraphicallyFinding an Inverse Graphically
The Inverse Reflection PrincipleThe points (a, b) and (b, a) in the coordinate plane are symmetricwith respect to the line y = x. The points (a, b) and (b, a) arereflections of each other across the line y = x.
Finding an Inverse GraphicallyFinding an Inverse Graphically
f x 1f x
The graph of a function is shown. Is the function one-to-one?Sketch a graph of the inverse of the function.
Yes!!!Yes!!!y = x
And one more new tool:And one more new tool:The Inverse Composition RuleThe 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 twofunctions are inverses… observe…
More PracticeMore Practice
3 1f x x 3 1g x x
Show algebraically that the given functions are inverses.
f g x 33 1 1x 1 1x x
g f x 33 1 1x 3 3x x
More PracticeMore Practice
3f x x
Show that the given function has an inverse and find a rule forthat inverse. State any restrictions of the domains of thefunction and its inverse.
Check the graph Is f one-to-one?
2 3x y
3x y
3y x
2 3y x
3, 0y x
3, 0x y where
where
3, 0y x where
3, 0y x where
Let’s graph theinverse togetherwith the original
function…
Whiteboard Problems…Whiteboard Problems…Find a formula for . Give the domain of Find a formula for . Give the domain of , including any restrictions “inherited” from , including any restrictions “inherited” from f.f.
1( )f x 1f
( ) 2 5f x x
3( ) 2f x x
1 1 5( )
2 2f x x
1 3( ) 2f x x
: ( , )D
: ( , )D
Whiteboard problems…Whiteboard problems…Confirm that f and g are inverses by Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x.showing that f(g(x)) = x and g(f(x)) = x.
3( )
4( ) 4 3
xf x
g x x