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Last edited 4/14/2009 10:28:00 AM

Image and inverse image

I have not written this part.  For now, read the Wikipedia article, which is pretty good .

The image of a function

[ label: imagfunc] 

If F:A→B is a function, it can easily happen that not every element of B is a value of F. For example, the function x‖→ x2 :R→R takes only nonnegative values.

Definition:image of a function

[ label: imdef] The image of F:A→B is the set of all values of F, in other words the set {b∈B‖∃a:A(F(a)=b)}. The image of F is also denoted ℑ(F).

Fact

[ label: T] his definition gives the equivalence:

Fact

[ label: F] or any function F, ℑ(F)⊆ cod F.

Usage

Many authors use the word "range" for the image, but others use "range" for the codomain.

Example

The image of the squaring function x‖→ x2 :R→R is the set of nonnegative real numbers.

Example

[ label: arbfunc]  Let the function F:{1,2,3}→{2,4,5,6} be defined by F(1)=4 and F(2)=F(3)=5. Then F has image {4,5}.

Remark

The image of a function can be difficult to determine if it is given by a formula; for example it requires a certain amount of analytic geometry (or calculus) to determine that the image of the function G(x)= x2 +2x+5 is the set of real numbers ≥4, and determining the image of more complicated functions can be very difficult indeed.

Exercise

Find the image of the function n‖→n+1:N→N. Answer: The set of positive integers.

Exercise

Find the image of the function n‖→n-1:Z→Z.

Exercise

Find the image of the function x‖→ x2 -1:R→R.

Exercise

Find the image of the function x‖→ x2 +x+1:R→R.

The image of a subset of the domain

The word "image" is used in a more general way which actually makes the image a function itself.

Definition:Image of a subset

[ label: subsetimage] Let F:A→B is a function, and suppose C⊆A. Then F(C) denotes the set {F(x)‖x∈C}, and is called the image of C under F. The map C‖→F(C) defines a function from PA to PB called the image function of F.

Remark

In particular, F(A) is what we called ℑ(F) in Chapter  [imagfunc] .

Example

[ label: imfuncex]  If F:{1,2,3}→{2,4,5,6} is defined as in  [arbfunc] by F(1)=4 and F(2)=F(3)=5, then F({1,2})={4,5} and F(∅)=∅. Thus the image of {1,2} under F is {4,5}.

Warning

The image function is not usually distinguished from F in notation. A few texts use F* :PA→PB, and so would write F(x) for x∈A but F* (C) for a subset C⊆A. In this text, as in almost all mathematics texts, we simply write F(C). Context usually disambiguates this notation (but there are exceptions!).

Exercise

Describe a function where our notation F(C) is ambiguous.

Exercise

Let F be defined as in Example  [imfuncex] . What are F({2,3}) and F({3})? Answer: F({2,3}={5} and F({3}) is also {5}.

Exercise

Let F:R→R be defined by F(x)= x2 +1. What is F((3…4))? What is F([-1…1])?

Exercise

Let F be defined as in Example  [imfuncex] . How many ordered pairs are in the graph of the image function of F?

Inverse images

Definition:Inverse image

[ label: invimdef] Let F:A→B be a function. For any subset C⊆B, the set

is called the inverse image of C under F, also written F-1 (C).

Example

Let F:{1,2,3}→{2,4,5,6} be defined (as in Example  [arbfunc] ) by F(1)=4 and F(2)=F(3)=5. Then F-1 ({4,6})={1}, F-1 ({5})={2,3}, and F-1 ({2,6})=∅.

Example

[ label: sqplusone]  For the function F:R→R defined by F(x)= x2 +1,

and

Inverse image as function

Like the image function, this inverse image function can also be defined as a function F-1 :PB→PA (note the reversal), where

for any D⊆B. F-1 is sometimes denoted F* .

Usage

It is quite common to write F-1 (x) instead of F-1 ({x}).

Example

For the function of Example  [sqplusone] , F-1 (3)={-2,2)}.

Exercise

Let F:R→R be defined by F(x)= x2 +1. What is F-1 ({1,2})? What is F-1 ((1…2))?

Exercise

For any function F:A→B, what is F-1 (∅)? What is F-1 (B)?