05a Cpt 5 Lecture Notes F16 - University of...

L32-Wed-16-Nov-2016-Sec-5-1-Composites-HW32-5-2-Inverses-HW33-Q26 page 1

L32-Wed-16-Nov-2016-Sec-5-1-Composite-HW32-5-2-Inverse-HW33-Q26

Be sure to go over the key to the exam and try and figure out what you did wrong and, more importantly, WHY you did it.


•

2 1 1 12 3 3 2 3 22 3

x xh x f g x f g x fxx x x

xx

D:g: x ≠ 0

D:f: x ≠ -3

D:h:

3 2 03 2

23

xx

x

Since g is the input for f we could also look at what value of x will make g = -3:

2

23

23

g xx

xx

This is the same result as above. So, domain of f g x is

2| 0,3

x x x

•

1 2 32 2 63 11

3

xj x g f x g f x g xx

x

D:g: x ≠ 0

D:f: x ≠ -3

D k: all reals So, domain of g f x is | 3x x


2 2 2 2 2

2

3 2 7 3 4 4 7 12 12 3 7

f g x f g x f x a

x a x ax a x ax a

Since f g x crosses y axis at 68, the point (0, 68) satisfies this equation:

2 2

2 2

2

12 12 3 7

68 12 0 12 0 3 775 3

5

f g x x ax a

a aa

a

2

2

12 60 68

12 60 68

f g x x xor

f g x x x


Sometimes, we can "decompose" a function. That is, we are given a function that is itself a composition and we find two functions that could make up the composition.

That is, find two functions f and g such that f g x h x

There are two operations here: subtracting 5 from x and squaring.

So, let's define

5g x x and 2f x x . We can see that

2

5 5h x x f x f g x f g x


5.2 One-to-One Functions & Inverse Functions

Defn: A function is a relation where each input corresponds to exactly one output.

For example, these are all functions (they pass the Vertical Line Test).


Which of the above are one to one functions?

ANS 3f x x and 1f xx


Note that if we restrict 2, 0f x x x then it is one-to-one.


This function maps 2 onto 3, 5 onto 7, and 8 onto -1.

The inverse does this mapping backwards.

That is, it maps 3 onto 2, 7 onto 5, and -1 onto 8.

The inverse can be written as 1 3,2 , 7,5 , 1,8f x .

From this example, we can see that the domain of f x is the range of 1f x AND

the range of f x is the domain of 1f x .


The inverse, 1

3xf x says to take the input and divide it by 3. This "undoes" multiplication by 3.

Since a function and its inverse "undo" each other, their composition results in the input. That is,

1 1f f x f f x x

For example, if 3f x x and 1

3xf x then

1 3

3 3x xf f x f x and 1 1 33

3xf f x f x x


2 3 3 3 332 2 32 32

xf g x f xx x xx

xx x

where x ≠ 0 and

•

3 6 2 6 6 32 3 32 2 23 3 22

32 2 33 322 2

x xx x x x xxg f x g x

x xxx x

where x ≠ 2 or 0


3

3

3

1 1 34

f x x

f

Thus, the point (1,4) satisfies

3 3f x x .

Therefore, the point (4, 1) will satisfy

1 3 3f x x

1 3

1 3

3

4 4 31

f x x

f


This means that the graph of a function and its inverse are symmetric with respect to the line y = x.

We can show this by showing that the line connecting (1, 4) and 4, 1) is perpendicular to the line y = x (i.e., its slope is -1), and the distance from (1,4) to the line y = x and the distance from (4, 1) to the line y = x are equal. So, if we are given the graph of a function we can graph its inverse by choosing points that are a reflection about the line y = x.


If we are given the equation of a function we can find its inverse by simply exchanging the x and y values and then solving for y.

Exchange x and y and then solve for y:

3

1 3

3

3

: 1: 1

11

f x y xf x x y

x yx y

D: all reals

R: all reals


Exchange x and y and then solve for y:

1

4:2

4:2

42

4 2

f x yx

f x xy

yx

yx

Df: x ≠ –2

D 1f x : x ≠ 0

Rf: y ≠ 0

R 1f x : y ≠ –2


1

2 3:4

2 3:4

4 2 34 2 32 4 32 4 3

4 32

xf x yxyf x x

yx y yxy x yxy y x

y x xxy

x

Df: x ≠ –4

D 1f x : x ≠ 2

Rf: y ≠ 2

R 1f x : y ≠ –4

05a Cpt 5 Lecture Notes F16 - University of...

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