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[email protected] • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§9.2b§9.2bInverse FcnsInverse Fcns



Review §Review §

Any QUESTIONS About• §9.2 → Composite Functions

Any QUESTIONS About HomeWork• §9.2 → HW-43

9.2 MTH 55



Inverse & One-to-One FunctionsInverse & One-to-One Functions

Let’s view the following two functions as relations, or correspondences:

Maine 1

Illinois 7

Iowa 2

Ohio 3

Domain Range

(inputs) (outputs)

States

ball Ann

rope Jim

phone Jack

car

Domain Range

(inputs) (outputs)

Toys




Suppose we reverse the arrows. We obtain what is called the inverse relation. Are these inverse relations functions?

Maine 1

Illinois 7

Iowa 2

Ohio 3

Range Domain

(inputs) (outputs)

States

ball Ann

rope Jim

phone Jack

Car

Range Domain

(inputs) (outputs)

Toys




Maine 1

Illinois 7

Iowa 2

Ohio 3

Range Domain

(inputs) (outputs)

States

ball Ann

rope Jim

phone Jack

Car

Range Domain

(inputs) (outputs)

Toys

Recall that for each input, a function provides exactly one output. The inverse of “States” correspondence IS a function, but the inverse of “Toys” is NOT.



One-to-One for “States” FcnOne-to-One for “States” Fcn

In the States function, different inputs have different outputs, so it is a one-to-one function.

In the Toys function, rope and phone are both paired with Jim.

Thus the Toy function is NOT one-to-one.

ball Ann

rope Jim

phone Jack

Car

Range Domain

(inputs) (outputs)

Toys



One-to-One SummarizedOne-to-One Summarized

A function f is one-to-one if different inputs have different outputs. That is, if for a and b in the domain of f with a ≠ b we have f(a) ≠ f(b) then the function f is one-to-one.

If a function is one-to-one, then its INVERSE correspondence is ALSO a FUNCTION.



One-to-One Fcn GraphicallyOne-to-One Fcn Graphically

Each y-value in the range corresponds to only one x-value in the domain

• i.e.; Each x has a Unique y



NOT a One-to-One FcnNOT a One-to-One Fcn

The y-value y2 in the range corresponds to TWO x-values, x2 and x3, in the domain.



NOT a Function at AllNOT a Function at All

The x-value x2 in the domain corresponds to the TWO y-values, y2 and y3, in the range.



Definition of Inverse FunctionDefinition of Inverse Function

Let f represent a one-to-one function. The inverse of f is also a function, called the inverse function of f, and is denoted by f−1.

If (x, y) is an ordered pair of f, then (y, x) is an ordered pair of f−1, and we write x = f−1(y). We have y = f (x) if and only if f−1(y) = x.



Example Example ff-values ↔ -values ↔ ff-1-1-values-values

Assume that f is a one-to-one function.a. If f(3) = 5, find f-1(5)

b. If f-1(−1) = 7, find f(7)

Solution: Recall that y = f(x) if and only if f-1(y) = x

a. Let x = 3 and y = 5. Now 5 = f(3) if and only if f−1(5) = 3. Thus, f−1(5) = 3.

b. Let y = −1 and x = 7. Now, f−1(−1) = 7 if and only if f(7) = −1. Thus, f (7) = −1.



Inverse Function PropertyInverse Function Property

Let f denote a one-to-one function. Then

f o f 1 x f f 1 x x

for every x in the domain of f–1.

1.

f 1 o f x f 1 f x x

for every x in the domain of f .

2.



Example Example Inverse Fcn Property Inverse Fcn Property

Let f(x) = x3 + 1. Show that 1 3( ) 1.f x x

Soln: 1 1( ) ( )f f x f f x

1 1( ) ( )f f x f f x

3 1f x

33 1 1x

1 1x x

1 3 1f x

33 ( 1) 1x

3 3x x



UNIQUE Inverse Fcn PropertyUNIQUE Inverse Fcn Property

Let f denote a one-to-one function. Then if g is any function such that

g = f –1. That is, g is the inverse function of f.

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, then



Verify Inverse FunctionsVerify Inverse Functions

Verify that the following pairs of functions are inverses of each other:

f x 2x 3 and g x x 3

2.

Solution: From the composition of f & g.

f og x f g x fx 3

2

2x 3

2

3 x 3 3

x



Verify Inverse FunctionsVerify Inverse Functions

Solution (cont.): Now Find

Observe:

g f x .

g o f x g f x g 2x 3

2x 3 3

2x

f g x g f x x,

This Verifies that f and g are indeed inverses of each other.



Example Example Find Inverse of a Fcn Find Inverse of a Fcn

Given that f(x) = 5x − 2 is one-to-one, then find an equation for its inverse

Solution: f (x) = 5x – 2

y = 5x – 2

x = 5y – 2

2

5

xy

1 2( )

5

xf x

Replace f(x) with y

Interchange x and y

Solve for y

Replace y with f-1(x)



Procedure for finding Procedure for finding ff−−11

1. Replace f(x) by y in the equation for f(x).

2. Interchange x and y.

3. Solve the equation in Step 2 for y.

4. Replace y with f−1(x).



Example Example Find the Inverse Find the Inverse

Find the inverse of the one-to-one function

Solution: y x 1

x 2Step 1

x y 1

y 2Step 2

Step 3 12

12

yxxy

yyx



Example Example Find the Inverse Find the Inverse

Step 3(cont.)

f 1 x 2x 1

x 1, x 1Step 4

yxyyxxxy 2122



Example Example Find Domain & Range Find Domain & Range

Find the Domain &Range of the function

Solution: Domain of f, all real numbers x such that x ≠ 2, in interval notation (−∞, 2)U(2, −∞)

Range of f is the domain of f−1 f 1 x 2x 1

x 1, x 1

Range of f is (−∞, 1) U (1, −∞)



Inverse Function MachineInverse Function Machine

Let’s consider inverses of functions in terms of function machines. Suppose that a one-to-one function f, has been programmed into a machine.

If the machine has a reverse switch, when the switch is thrown, the machine performs the inverse function, f−1. Inputs then enter at the opposite end, and the entire process is reversed.



Reverse Switch GraphicallyReverse Switch Graphically

Forward

Reverse



Horizontal Line TestHorizontal Line Test Recall that to be a

Function an (x,y) relation must pass the VERTICAL LINE test

In order for a function to have an inverse that is a function, it must pass the HORIZONTAL-LINE test as well

NOT a Function – Fails the Vertical Line Test



Horizontal Line Test DefinedHorizontal Line Test Defined If it is impossible to

draw a horizontal line that intersects a function’s graph more than once, then the function isone-to-one.

For every one-to-one function, an inverse function exists.

A Function withOUT and Inverse – Fails the Horizontal Line Test (not 1-to-1)



Example Example Horizontal Line Test Horizontal Line Test Determine whether the function f(x) = x2 + 1 is

one-to-one and thus has an inverse fcn. The graph of f is shown. Many

horizontal lines cross the graph more than once. For example, the line y = 2 crosses where the first coordinates are 1 and −1. Although they have different inputs, they have the same output: f(−1) = 2 = f(1). The function is NOT one-to-one, therefore NO inverse function exists

x

y

-5 -4 -3 -2 -1 1 2 3 4 5

4

3

6

2

5

1

-1

-2

78



Example Example Horizontal Ln Test Horizontal Ln Test

Use the horizontal-line test to determine which of the following fcns are 1-to-1

a. b.

Soln a. • No horizontal line

intersects the graph of f in more than one point, therefore the function f is one-to-one



Example Example Horizontal Ln Test Horizontal Ln Test

Use the horizontal-line test to determine which of the following fcns are 1-to-1

a. b.

Soln b. • No horizontal line

intersects the graph of f in more than one point, therefore the function f is 1-to-1



Graphing Fcns and Their InversesGraphing Fcns and Their Inverses

How do the graphs of a function and its inverse compare?



Example Example Graphs Inverse Graphs Inverse FcnFcn Graph f(x) = 5x − 2 and f−1(x) = (x + 2)/5

on the same set of axes and compare Solution:

x -5 -4 -3 -2 -1 1 2 3 4 5

-3

2

-2

3

-1

1

6

54

-4

-5

f (x) = 5x – 2

f -1(x) = (x + 2)/5

Note that the graph of f−1(x) can be drawn by reflecting the graph of f across the line y = x.

When x and y are interchanged to find a formula for f−1(x), we are, in effect, Reflecting or Flipping the graph of f.



Visualizing Inverses Visualizing Inverses

The graph of f−1 is a REFLECTION of the graph of f across the line y = x.



Example Example Use Use yy = = xx Mirror Ln Mirror Ln

The graph of the function f is shown at Lower Right. Sketch the graph of the f−1

Soln



Example Example Inverse or Not? Inverse or Not? Ray’s Music Mart has six employees. The

first table lists the first names and the Social Security numbers of the employees, and the second table lists the first names and the ages of the employees

a. Find the inverse of the function defined by the first table, and determine whether the inverse relation is a function

b. Find the inverse of the function defined by the second table, and determine whether the inverse relation is a function



Example Example Inverse or Not? Inverse or Not?

Dwayne 590-56-4932

Sophia 599-23-1746

Desmonde 264-31-4958

Carl 432-77-6602

Anna 195-37-4165

Sal 543-71-8026

Solution:Every y-value corresponds to exactly one x-value. Thus the inverse of the function defined in this table is a function



Example Example Inverse or Not? Inverse or Not?

Solution:There is more than one x-value that corresponds to a y-value. For example, the age of 24 yields the names Dwayne and Anna. Thus the inverse of the function defined in this table is NOT a function.

Dwayne 24

Sophia 26

Desmonde 42

Carl 51

Anna 24

Sal 26



Example Example Hydrostatic Pressure Hydrostatic Pressure

The formula for finding the water pressure p (in pounds per square inch, or psi),at a depth d (in feet) below the surface is

A pressure gauge on a Diving Bell breaks and shows a reading of 1800 psi. Determine how far below the surface the bell was when the gauge failed



Example Example HydroStatic P HydroStatic P

Solution: The depth is given by the inverse of

Solve theInverseEqn for p

p 15d

3333p 15d

d 33p

15

Let p = 1800 psi

d 33 1800

15d 3960

The Diving Bell was 3960 feet below the surface when the gauge failed



WhiteBoard WorkWhiteBoard Work

Problems From §9.2 Exercise Set• 38, 42, 60, 68, 76

Some Temperature Scales



All Done for TodayAll Done for Today

Old StyleDiving

Bell



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

AppendiAppendixx

–

srsrsr 22

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