One-To-One and Inverse Functions

7/24/2019 One-To-One and Inverse Functions

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One-to-one and InverseFunctions

Digital Lesson


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A functiony =f(x) with domain D is one-to-oneon D

if and only if for everyx1

andx

in D! f(x1

) =f(x

) im"lie

thatx1=x#

Afunctionis a ma""ing from its domain to its range

so that each element!x! of the domainis ma""ed to one!and only one! element!f(x)! of the range#

Afunction isone-to-oneif each elementf(x) of therangeis ma""edfromone! and only one! element!x!

of the domain#


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x

y

Horizontal Line Test

A functiony =f(x) is one-to-oneif and only ifno hori$ontal line intersects the gra"h ofy =f(x)

in more than one "oint#

y = %

Example& 'he function

y = xx + % is not one-to-one

on the real num*ers *ecause the

liney = % intersects the gra"h at

*oth (+! %) and (! %)#

(+! %) (! %)


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one-to-one

Example& A""ly the horizontal line testto the gra"hs

*elow to determine if the functions are one-to-one#

a)y =x, *)y = x,+ ,xx 1

not one-to-one

x

y

-

x

y

-


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Domain.ange

Inverse relationx =|y| + 1

2

1

0

-1-2

x y

3

2

1

Domain .ange

2

1

0

-1-2

x y

3

2

1

Functiony =|x| + 1

/very functiony =f(x) has an inverse relationx =f(y)#

'he ordered "airs of &

y=|x| + 1 are 0(-! ,)! (-1! )! (+! 1)! (1! )! (! ,)#

x =|y| + 1 are 0(,! -)! (! -1)! (1! +)! (! 1)! (,! )#

'he inverse relation is not a function# It "airs to *oth -1 and 21#


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'he ordered "airs of the functionfare reversedto

"roduce the ordered "airs of the inverse relation#

Example& 3iven the function

f =0(1! 1)! (! ,)! (,! 1)! (! )! its domain is 01! ! ,!

and its range is 01! ! ,#

'he inverse relation off is0(1! 1)! (,! )! (1! ,)! (! )#

'he domainof the inverse relation is the rangeof the

original function#

'herangeof the inverse relation is thedomainof the

original function#


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y =x

'he gra"hs of a relation and its inverse are reflections

in the liney=x#

'he ordered "airs offaregiven *y the e4uation #

)(,

=x

y

)( , =

yx

)( , =

xy

Example&Find the gra"h of the inverse relation

geometricallyfrom the gra"h off(x)=

)( , x

x

y

-

-

)(

,

= yx

'he ordered "airs of the inverse are

given *y #


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Example&Find the inverse relation algebraicallyfor the

functionf(x) = ,x2 #

y= ,x2 Original e4uation definingf

x= ,y2 5witchxandy#

,y2 =x .everse sides of the e4uation#

'o calculate a value for the inverse off!subtract 2, thendivide by 3#

y= 5olve for y#,

)( x

'o find the inverse of a relation algebraically!

interchangexandyand solve fory#


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For a functiony=f(x)! the inverse relation off

is afunctionif and only if f isone-to-one#

For a functiony=f(x)! the inverse relation off

is afunctionif and only if the gra"h of f "asses the

horizontal line test#

If fis one-to-one! the inverse relation of f

is a function called theinverse function off#

'he inverse function ofy=f(x) is writteny=f -1(x)#


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y =f(x)

y =xy =f -1(x)

Example&From the gra"h of the functiony=f(x)!

determine if the inverse relation is a function and! if it

is! s6etch its gra"h#

'he gra"h off"asses

the hori$ontal line test#

'he inverse

relation is a function#

.eflect the gra"h offin the liney=x to "roduce the

gra"h of f -1#

x

y


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'he inverse function is an 7inverse8 with res"ect to

the o"eration of composition of functions#

'he inverse function 7undoes8 the function!

that is!f -1( f(x)) =x#

'he function is the inverse of its inverse function!

that is!f ( f -1(x)) =x#

Example& 'he inverse off(x) =x,

is f-1

(x) = #x,

f -1(f(x)) = =xandf (f -1(x)) = ( ),=x#,

x,

x,


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It follows thatg =f -1#

Example&9erify that the functiong(x) =is the inverseoff(x) = x 1#

fg(x)) = g(x) 1 = ( ) 1 = (x2 1) 1 =x

1+x

1+x

g(f(x)) = = = =x

)1)1(( +x

x

)1)(( +xf

One-To-One and Inverse Functions

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