Inverse Functions Undoing What Functions Do. 6/1/2013 Inverse Functions 2 One-to-One Functions...

Inverse Functions

Undoing What Functions Do

Inverse Functions 26/1/2013

Inverse Functions

One-to-One Functions Definition

A function f is a one-to-one function if no two ordered pairs of f have the same second component

Note: One-to-one is often written as 1-1


One-to-One Functions 1-1 Examples:

1. f = { (1, 3), (2, 5), (3, 2), (7, 1) } 2. g = { (1, 3), (2, 5), (3, 6), (7, 3) } 3. h = { (5, 3), (2, 9), (5, 6), (8, 7) } 4. f(x) = (x – 4)2 + 7 5. g(x) = x + 1 6. f(x) = |x + 1|

Inverse Functions

NOT 1-1

NOT 1-1

NOT 1-1

NOT a function

WHY ?

WHY ?


Inverse Functions

Horizontal Line Test No horizontal line intersects the graph

of a 1-1 function more than once

Examples

x

y(x)

x

y(x)

● ●●1-1 function

Not 1-1


Inverse Functions

Horizontal Line Test No horizontal line intersects the graph

of a 1-1 function more than once

More Examples

x

y(x)

x

y(x) x

y(x)

● ●

●

●●●

●●

NOT a function !

Not 1-1 Not 1-1


Inverse Functions

The Inverse of a Function Example:

Let y = f(x) = 2x + 1

Solving for x : x = (1/2)(y – 1) = g(y)

Each x is mapped to a particular y by f(x), and that y is mapped back to the original x by g(y)

So, if x = 3 , then f(3) = 2(3) + 1 = 7 and g(7) = (1/2)(7 – 1) = 3


Inverse Functions

The Inverse of a Function If x = 3 , then

f(3) = 2(3) + 1 = 7

3 7

f

g

and g(7) = (1/2)(7 – 1) = 3

Domain f

Domain g

Questions:What are

g is the inverse of f

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

(f g)(y) = f(g(y)) ?


Inverse Notation If function g is the inverse function for

function f we write this as g = f–1

Note:

f–1 does NOT mean the reciprocal of f

That is f–1(x) ≠

Inverse Functions

f(x)1


Inverse Notation If function g is the inverse function for

function f we write this as g = f –1

Questions

Inverse Functions

What is (f f –1)(x) ?

(f –1 f)(x) ?What is

Does every function have an inverse ?If not, what guarantees an inverse ?


Definition

A 1-1 function f has inverse f –1

Inverse Functions

(f –1 f)(x) = f

–1(f(x)) = xfor every x in the domain of f

for every x in the domain of f –1

(f f –1)(x) = f(f

–1(x)) = xAND

IF and ONLY IF


Definition A 1-1 function f has inverse f –1

Inverse Functions

(f –1 f)(x) = f

–1(f(x)) = x

for every x in the domain of f

for every x in the domain of f –1

Note:The name of variable x is a dummy name

… both can use xSince f and f –1 are different functions

IF and ONLY IF

AND(f f

–1)(x) = f(f –1(x)) = x


Inverse Functions Example: For f(x) = 2x + 1 and f–1(x) = (1/2)(x – 1) we have

(f–1 f)(x)

= f–1(2x + 1)

= (1/2)((2x + 1) – 1)

= x

x

y

f(x)

= 2

x +

1

f–1 (x) = (x – 1)/2

L1

L2

= f–1(f(x))


Inverse Functions Example: f(x) = 2x + 1 and f–1(x) = (1/2)(x – 1)

k

2k + 1

2k + 1

k

y = x

(2k + 1 , k)

(k , 2k + 1)

x

y

f(x)

= 2

x +

1

f–1 (x) = (x – 1)/2

L1

L2

● ●

●

●

●

●

Each point on L2

... and converselycorresponding point on L1

Pick an x = k… and follow it

Feed 2k + 1 to f –1

… to return k

is a reflection across line y = x of a


Inverse Functions and Graphs

For any 1-1 function f(x)

with graph { (x, y) y = f(x) }

the inverse function f–1

has a graph { (y, x) x = f–1(y) }

which is a reflection

of the graph of f(x)

through the line y = x

x

y

f(k)

k

k

f(k)

y = f(x)

y = f–1(x)

y = x

(k, f(k))

(f(k), k)

●

● ●

●

●

●


x

y

f(k)

k

k

f(k)

y = f(x)

y = f–1(x)

y = x

(k, f(k))

(f(k), k)

●

● ●

●

●

●

Inverse Functions and Graphs 1-1 function f(x) with inverse function f–1

Note that y = f(x) so we have x = f–1(y)

Interchanging the names of x and y yields the graph of f–1 as { (x, y) y = f–1(x) }


Inverse Functions Inverse Function Fact

A function f has an inverse function f –1

IF and ONLY IF

Example

1. f = { (3, 15), (4, 10), (5, 7), (6, 4), (7, 3) }

f–1 = { (15, 3), (10, 4), (7, 5), (4, 6), (3, 7) }

(f –1 f)(4) f

–1(f(4)) = f –1(10) = 4 =

f is a 1-1 function

Thus


Inverse Functions Example

1. f = { (3, 15), (4, 10), (5, 7), (6, 4), (7, 3) }

f–1 = { (15, 3), (10, 4), (7, 5), (4, 6), (3, 7) }

In General :

(f f –1)(4) ?What about

Does 4 have to be in the domain of f –1 ?

=(f –1 f)(4) f –1(f(4)) = f –1(10) 4 =

Thus

Question:

(f –1 f)(x) f

–1(f(x)) = x =



2.

(f f –1)(2) ?What about

f = { (x, y) | y = }x – 4 + 7

… for x ≥ 4

Is this 1-1 ?

Well …

… is it ? NOTE:

… if y ≥ 7 ?

… so that y ≥ 7

… 2 would have to be in the range of f

f –1 = { (y, x) | x = (y – 7)2 + 4 }

(f –1 f)(4) f

–1(f(4)) = 4 =

… if y < 7 ?

f(x) ≥ 7 > 2 for all x ≥ 4



2.

f = { (x, y) | y = }x – 4 + 7

f –1 = { (y, x) | x = (y – 7)2 + 4 }

Question:What are the domain and range of f ?

What about f –1 ?

Dom f = { x x ≥ 4 } = [ 4, )Range f = { x x ≥ 7 } = [ 7, )

Dom f–1 = { x x ≥ 7 } = [ 7, )Range f–1 = { x x ≥ 4 } = [ 4, )


Examples: Find the inverses where they exist

3. g(x) =

Inverse Functions

x + 5 + 1 Domain ?

g –1(x) =

Range ? [ 1, ) [ –5, )

(x – 1)2 – 5 Domain ?Range ? [ –5 , )

[ 1, )


Examples: Find the inverses where they exist

4. h(t)

5. y = | x + 1 |

Inverse Functions

Domain ?

Range ?h –1(t) =

( – , ) [ 6, ) NONE

12 t2 + 6=

h is not 1-1 so has no inverse

Domain ?

Range ?y–1(x) = NONE

( –, )

[ –1, )

y is not 1-1 so has no inverse


Inverse Functions

Example: 6. Find the graph of the inverse function if it exists Identify intercepts Plot y = x line Find intercept reflections

x

y

(0,–5)

(2, 0)

y = f(x)

(–5, 0)

(0, 2)

y = x


Inverse Functions

Example: 6. Find the graph of the inverse function Plot the inverse graph

x

y

(0,–5)

(2, 0)

y = f(x)

(–5, 0)

(0, 2)y = f –1(x)

y = x

Question:

What are the equations of the graphs of f and f–1 ?


Inverse Functions

x

y

(0,–5)

(2, 0)

y = f(x)

(–5, 0)

(0, 2)y = f –1(x)

y = x

Question:


52y = f(x) = x – 5

Use intercepts for slopes:

y = f –1(x) = x + 2

25


Inverse Functions

Example: 7. Find the graph of the inverse function if it exists Identify intercepts Plot y = x line Find intercept reflections

x

y

y = f(x)

(3, 0)

(0, 3)

y = x

(0, 6)

(6, 0)


Inverse Functions

Example: 7. Find the graph of the inverse function Plot the inverse graph x

y

y = f(x)

(3, 0)

(0, 3)

y = f –1(x)

y = x

(0, 6)

(6, 0)

Question:



Inverse Functions

x

y

y = f(x)

(3, 0)

(0, 3)

y = f –1(x)

y = x

(0, 6)

(6, 0)

Question:


Use intercepts for slopes:

y = f –1(x) = –2x + 6

y = f(x) = x + 3 12

–


Think about it !

Inverse Functions Undoing What Functions Do. 6/1/2013 Inverse Functions 2 One-to-One Functions...

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