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Algebra 2: Algebra 2: Section 7.4Section 7.4Inverse FunctionsInverse Functions

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Inverse RelationInverse Relation

Maps the output back to Maps the output back to original inputoriginal input

DomainDomain of inverse is the of inverse is the rangerange of the original of the original functionfunction

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Finding Inverse of a Finding Inverse of a FunctionFunction

(Algebraically)(Algebraically) Rewrite function name;Rewrite function name;

usually usually f(x)f(x) as as yy

Switch the Switch the xx’s and ’s and yy’s’s

Solve for ySolve for y

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Finding Inverse of a Finding Inverse of a FunctionFunction

3 6y x 3 6x y

12

3x y

12

3y x

Solve for y

Switch x and y

6

3

xy

6 3x y

Don't have an f(x)

to change into y!

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Power FunctionPower Function

by ax A function of the formA function of the form

WhereWhereaa is a real number is a real numberbb is rational is rational

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Finding Inverse of a Finding Inverse of a FunctionFunction5( )f x x

5y x5x y

5 x y 5y x

Switch x and y

Solve for y

Replace f(x) with y

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Finding Inverse of a Finding Inverse of a FunctionFunction

22 4y x 22 4x y

24 2x y

24

2

xy

4

2

xy

4

2

xy

Switch x and y

Solve for y

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Verifying Functions are Verifying Functions are InversesInverses

Functions are inverses of Functions are inverses of each other if…each other if… f(g(x)) = xf(g(x)) = xANDANDg(f(x)) = xg(f(x)) = x

f f -1-1(x) = g(x)(x) = g(x)Reads “inverse of Reads “inverse of f”f”

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Verify that Verify that f f and and gg are are inversesinverses 1

( ) 3 6; ( ) 23

f x x g x x

( ( )) 6 6f g x x

( ( ))f g x x

f and g are inverses of each other!!!

( ( )) 2 2g f x x ( ( ))g f x x

( ( )) ( ( )) ......ANBecause f g x Dx g f x x

( ( ))f g x 13( 2) 6

3x

( ( ))g f x 1( 3 6) 2

3x

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HomeworkHomework P.426P.426

#16-24 all#16-24 all

#25-31 odd#25-31 odd

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Algebra 2: Algebra 2: Section 7.4Section 7.4Inverse FunctionsInverse Functions

(Day 2)(Day 2)

Warm-UpWarm-Up

What is the range of a function?What is the range of a function?Output valuesOutput valuesyy-values-values

What is the domain of a What is the domain of a function?function? Input valuesInput valuesxx-values-values

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Finding Inverses of a Finding Inverses of a FunctionFunction

(Graphically)(Graphically) To create the inverse graph of a To create the inverse graph of a

function…function… Reflect the original graph Reflect the original graph

across the line across the line y = xy = x.. Examples on Board!!!Examples on Board!!!

Swap x and y values of plotted Swap x and y values of plotted pointspointsy = xy = x22 y = xy = x33

Sketchpad Example

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Warm-UpWarm-Up Write the image of each Write the image of each

point after its reflection in point after its reflection in the line the line y = xy = x..

1.1. (2, 3)(2, 3) 2.2. (-2, 4)(-2, 4) 3.3. (-1, -1)(-1, -1) 4.4. (1, -3)(1, -3)

(3, 2)

(4, -2)

(-1, -1)

(-3, 1)

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Inverse FunctionsInverse Functions An inverse relation may or may An inverse relation may or may

not be a function not be a function (even if the (even if the original IS a function!)original IS a function!)

Graph original in calculator Graph original in calculator (y(y11))

Graph inverse in calculator Graph inverse in calculator (y(y22)) Is the inverse a function?Is the inverse a function?

How could you tell by looking at only the How could you tell by looking at only the graph of the original function?graph of the original function?

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Horizontal Line TestHorizontal Line Test Look at original functionLook at original function

If no horizontal line intersects the graph If no horizontal line intersects the graph of the function more than once, thenof the function more than once, then

the inverse of the original will also be a the inverse of the original will also be a functionfunction

So, if a relation passes the vertical So, if a relation passes the vertical and horizontal line tests then the and horizontal line tests then the original relation and its inverse are original relation and its inverse are functionsfunctions

Vertical Line Test Vertical Line Test original original functionfunction

Horizontal Line Test Horizontal Line Test inverse inverse functionfunction

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Examples: Find the Examples: Find the inverseinverse

81. ( ) 256f x x

8256y x

8256x y8

256

xy

8

256

xy

8

8 256

xy

8

2

xy

: [0, )D

Re :

int ?

view what is the

domain in erval form

Re

0 0

striction of

x or x ?

How would that

change your answer

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HomeworkHomework P.427P.427

#36-56 all#36-56 all

Use graphing calculator for Use graphing calculator for #48-56#48-56 Draw sketch of graph on Draw sketch of graph on

homeworkhomework

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Verifying if Functions are Verifying if Functions are InversesInverses

Using TI-83Using TI-83 Enter each function into Y1 and Y2Enter each function into Y1 and Y2 Enter Y1(Y2) into Y3Enter Y1(Y2) into Y3 Enter Y2(Y1) into Y4Enter Y2(Y1) into Y4 Turn on graph of Y3 onlyTurn on graph of Y3 only

See if it is the graph of y=xSee if it is the graph of y=x Turn on graph of Y4 onlyTurn on graph of Y4 only

See if it is the graph of y=xSee if it is the graph of y=x

Verify #31, p.426 using TI-83Verify #31, p.426 using TI-83