1.6 Inverse Functions
Objectives
•Find inverse functions informally and verifythat two functions are inverse functions ofeach other.
•Determine from the graphs of functionswhether the functions have inversefunctions.
•Determine if functions are one to one.
•Find inverse functions algebraically.
Mapping Relations
•Mapping shows how each member of
the domain and range are paired.
1. 2.
–Example: 1943
-3210
7-5-2-6
7
–1. Relation: {(1,2), (-3,9), (7,-5)}
–2. Relation: {(4,-6), (1,3), (1,0), (-2,7)}
Mapping Example
1
5
7
-1
0
2
11
Relation
{(1, 0), (5, 2), (7, 2), (-1, 11)}
Domain
{1, 5, 7, -1}
Range
{0, 2, 11}
Types of Relations
Which Relations are also Functions?
•Many to One Relationship
•One to One Relationship
{ (3, 2), (1, 2), (2, 2), (8, 2), (7, 2) }
{ (0, 2), (1, 0), (2, 6), (8, 12) }
What is an Inverse Function?
INVERSE FUNCTION – reversing afunction, “undoing” it.
f -1 notates an inverse function. (not 1/f)
Set X
Set Y
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4
Remember we talked about functions---taking a set X andmapping into a Set Y
An inverse function would reverse that process and mapfrom SetY back into Set X
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Domain
Range
Inverse relation x = |y| + 1
210-1-2
x
y
321
Domain
Range
210-1-2
x
y
321
Function y = |x| + 1
Every function y = f (x) has an inverse relation x = f (y).
The ordered pairs of : y = |x| + 1 are {(-2, 3), (-1, 2), (0, 1), (1, 2), (2, 3)}.
x = |y| + 1 are {(3, -2), (2, -1), (1, 0), (2, 1), (3, 2)}.
The inverse relation is not a function. It pairs 2 to both -1 and +1.
Inverse Relation
Inverse Relations: Ordered Pairs
•Original relation, y1: {(0, 4) (2, 6) (5, 9) (10, 14)}
•Inverse relation, y2: {(4, 0) (6, 2) (9, 5) (14, 10)}
•To find the inverse relation, represented by orderedpairs, simply switch the x and y of each ordered pair.
•If y1 contains (x, y), then y2 contains (y, x).
•Are these two relations, a function (f(x)) and it’sinverse function (f-1(x))?
YES
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If we map what we get out of the function back, we won’t always have afunction going back.
Since going back, 6 goes back to both 3 and 5, the mapping goingback is NOT a function These functions are calledmany-to-one functions
Only functions that pair the y value (value in the range) with onlyone x will be functions going back the other way. These functionsare called one-to-one functions.
A function y = f (x) with domain D is one-to-one on Dif and only if for every x1 and x2 in D, f (x1) = f (x2) impliesthat x1 = x2.
A function is a mapping from its domain to its rangeso that each element, x, of the domain is mapped to one,and only one, element, f (x), of the range.
A function is one-to-one if each element f (x) of therange is mapped from one, and only one, element, x,of the domain.
One-to-One Functions
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This would not be a one-to-one function because to be one-to-one, each ywould only be used once with an x.
This function IS one-to-one. Each x is paired with only one y andeach y is paired with only one x
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Only one-to-one functions will have inverse functions, meaning themapping back to the original values is also a function.
Recall that to determine by the graph if an equation is a function, wehave the vertical line test.
If a vertical line intersects the graph of an equation more thanone time, the equation graphed is NOT a function.
This is a function
This is NOT afunction
This is a function
x
y
2
2
Horizontal Line Test
A function y = f (x) is one-to-one if and only ifno horizontal line intersects the graph of y = f (x)in more than one point.
y = 7
Example: The functiony = x2 – 4x + 7 is not one-to-oneon the real numbers because theline y = 7 intersects the graph atboth (0, 7) and (4, 7).
(0, 7)
(4, 7)
To be a one-to-one function, each y value could only be paired with one x. Let’slook at a couple of graphs.
Look at a y value (for example y =3)and see if there is only one xvalue on the graph for it.
This is a many-to-one function
For any y value, a horizontal line willonly intersection the graph once so willonly have one x value
This then IS a one-to-one function
If a horizontal line intersects the graph of an equation morethan one time, the equation graphed is NOT a one-to-onefunction and will NOT have an inverse function.
This is aone-to-one function
This is NOT a one-to-one function
This is NOT a one-to-one function
one-to-one
Example: Apply the horizontal line test to the graphsbelow to determine if the functions are one-to-one.
a) y = x3
b) y = x3 + 3x2 – x – 1
not one-to-one
x
y
-4
4
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8
x
y
-4
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8
Why are one-to-onefunctions important?
One-to-One Functions
have
Inverse functions
A function, f, hasan inverse function, g,if and only if (iff) the function fisa one-to-one (1-1) function.
Existence of an Inverse Function
The ordered pairs of the function f are reversed to produce theordered pairs of the inverse relation.
Example: Given the function f = {(1, 1), (2, 3), (3, 1), (4, 2)}, its domain is {1, 2, 3, 4} and itsrange is {1, 2, 3}.
The inverse relation of f is {(1, 1), (3, 2), (1, 3), (2, 4)}.
The domain of the inverse relation is the range of the originalfunction.
The range of the inverse relation is the domain of the originalfunction.
Ordered Pairs
Domain of f
Range of f
Domain and Range
•The domain of f is the range of f -1
•The range of f is the domain of f -1
•Thus ... we may be required to restrict thedomain of f so that f -1 is a function
Restricting a Domain
•When the inverse of a function is not afunction, the domain of the function canbe restricted to allow the inverse to be afunction.
•In such cases, it is convenient to consider“part” of the function by restricting thedomain of f(x). If the domain isrestricted, then its inverse is a function.
Restricting the Domain
Recall that if a function is not one-to-one,
then its inverse will not be a function.
Restricting the Domain
If we restrict the domain values of f(x) to those greater than or equal tozero, we see that f(x) is now one-to-one and its inverse is now afunction.
Your Turn:
(c)Suggest a suitable domain for so that the inverse
function can be found.
(a)Sketch the function where
.
(d)On the same axes sketch .
(b)Write down the range of .
(a)
Solution:
( We’ll look at the otherpossibility next. )
(c) Restricted domain:
(b) Range of :
(c)Suggest a suitable domain for so that the inverse
function can be found.
(a)Sketch the function where
.
(d)On the same axes sketch .
(b)Write down the range of .
The other possibility:
(a)
(c)
Suppose you chose
for the domain
(b) Range of :
(c)Suggest a suitable domain for so that the inverse
function can be found.
(a)Sketch the function where
.
(d)On the same axes sketch .
(b)Write down the range of .
y = x
The graphs of a relation and its inverse are reflectionsin the line y = x.
The ordered pairs of f aregiven by the equation .
Example: Find the graph of the inverse relationgeometrically from the graph of f (x) =
x
y
2
-2
-2
2
The ordered pairs of the inverse aregiven by .
Let’s consider the function and compute some values andgraph them.
Notice that the x and y values tradedplaces for the function and its inverse.
x f (x)
-2 -8-1 -1 0 0 1 1 2 8
Is this a one-to-one function?
Yes, so it will have an inversefunction
What will “undo” a cube?
A cube root
This means “inverse function”
x f -1(x)
-8 -2-1 -1 0 0 1 1 8 2
Let’s take thevalues we got outof the function andput them into theinverse functionand plot them
These functions arereflections of each otherabout the line y = x
(2,8)
(8,2)
(-8,-2)
(-2,-8)
Inverses of Functions
If the inverse of a function f is also afunction, it is named f 1 and read “f-inverse.”
The negative 1 in f 1 is not an exponent.This does not mean the reciprocal of f.
f 1(x) is not equal to
y = f(x)
y = x
y = f -1(x)
Example: From the graph of thefunction y = f (x), determine if theinverse relation is a function and, if itis, sketch its graph.
The graph of f passes thehorizontal line test.
The inverserelation is a function.
Reflect the graph of f in the line y = x to produce the graphof f -1.
x
y
Determine Inverse Function
Consider the graph of the function
The inverse function is
Example:
Consider the graph of the function
The inverse function is
An inverse function is just a rearrangement with x and y swapped.
So the graphs just swap x and y!
x
x
x
x
x
x
x
x
is a reflection of in the line y = x
What else do you notice about the graphs?
x
The function and its inverse must meet on y = x
Your Turn:Graph f(x) and f -1(x) on the same graph.
On the same axes, sketch the graph of
and its inverse.
x
Solution:
Your Turn:
Example: Find the inverse relation algebraically for thefunction f (x) = 3x + 2.
y = 3x + 2 Original equation defining f
x = 3y + 2 Switch x and y.
3y + 2 = x Reverse sides of the equation.
To calculate a value for the inverse of f, subtract 2, then divideby 3.
y = Solve for y.
To find the inverse of a relation algebraically, interchange x andy and solve for y.
Inverse Relation Algebraically
For a function y = f (x), the inverse relation of fis a function if and only if f is one-to-one.
For a function y = f (x), the inverse relation of fis a function if and only if the graph of f passes thehorizontal line test.
If f is one-to-one, the inverse relation of fis a function called the inverse function of f.
The inverse function of y = f (x) is written y = f -1(x).
Inverse Function
Steps for Finding the Inverse of aOne-to-One Function
Replace f(x)with y
Trade x and yplaces
Solve for y
y = f -1(x)
Find the inverse of
Replace f(x)with y
Trade x and yplaces
Solve for y
y = f -1(x)
Let’s check this by doing
or
Ensure f(x) is one to one first.Domain may need to be restricted.
Your Turn:Find the inverses of these functions:
1)
2)
3)
4)
5)
6)
So geometrically if a function and its inverse aregraphed, they are reflections about the line y = x and thex and y values have traded places. The domain of thefunction is the range of the inverse. The range of thefunction is the domain of the inverse. Also if we startwith an x and put it in the function and put the result inthe inverse function, we are back where we started from.
Given two functions, we can then tell if they are inversesof each other if we plug one into the other and it “undoes”the function. Remember subbing one function in theother was the composition function.
So if f and g are inverse functions, their compositionwould simply give x back. For inverse functions then:
A function, f, has an inversefunction, g, if and only iff(g(x)) = x and g(f(x)) = x,for every x in domain of gand in the domain of f.
Alternate Definition of an InverseFunction
The inverse function is an “inverse” with respect tothe operation of composition of functions.
The inverse function “undoes” the function,that is, f -1( f (x)) = x.
The function is the inverse of its inverse function,that is, f ( f -1(x)) = x.
Example: The inverse of f (x) = x3 is f -1(x)= .
3
f -1( f(x))= = x and f ( f -1(x)) =( )3 = x.
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3
3
Composition of Functions
Verify that the functions f and g are inverses of each other.
If we graph (x - 2)2 it is a parabola shifted right 2.
Is this a one-to-one function?
This would not be one-to-onebut they restricted the domainand are only taking thefunction where x is greaterthan or equal to 2 so we willhave a one-to-one function.
Verify that the functions f and g are inverses of each other.
Since both of these = x, if you start with x and apply thefunctions they “undo” each other and are inverses.
It follows that g = f -1.
Example: Verify that the function g(x) = is the inverse of f(x) = 2x – 1.
f(g(x)) = 2g(x) – 1 = 2( ) – 1 = (x + 1) – 1 = x
g( f(x)) = = = = x
Homework
•Section 1.6, pg. 69 – 72:
Vocabulary Check #1 – 5 all
Exercises: #1-33 odd, 47-55 odd, 111, 113