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# 12.6: Image and Preimage

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It is time to take up a matter of notation that you will encounter in future mathematics classes. Suppose we have a function $$f: A \rightarrow B$$. If $$X \subseteq A$$, the expression $$f(X)$$ has a special meaning. It stands for the set $$\{f(x) : x \in X\}$$. And if $$Y \subseteq B$$, then $$f^{-1}(Y)$$ has a meaning even if f is not invertible: it stands for the set $$\{x \in A : f(x) \in Y\}$$. Here are the precise definitions.

Definition 12.9

Suppose $$f : A \rightarrow B$$ is a function.

1. If $$X \subseteq A$$, the image of X is the set $$f(X) = \{f(x):x \in X \subseteq B\}$$.

2. If $$Y \subseteq B$$, the preimage of Y is the set $$f^{-1}(Y) = \{x \in A : f(x) \in Y \subseteq A\}$$.

In words, the image $$f(X)$$ of X is the set of all things in B that f sends elements of X to. (Roughly speaking, you might think of $$f(X)$$ as a kind of distorted “copy” or “image” of X in B.) The preimage $$f^{-1}(Y)$$ of Y is the set of all things in A that f sends into Y .

Maybe you have already encountered these ideas in linear algebra, in a setting involving a linear transformation $$T : V \rightarrow W$$ between two vector spaces. If $$X \subseteq V$$ is a subspace of V, then its image $$T(X)$$ is a subspace of W. If $$Y \subseteq W$$ is a subspace of W, then its preimage $$T^{-1}(Y)$$ is a subspace of V. (If this does not sound familiar, then ignore it.)

Example 12.13

Let $$f :\{s,t,u,v,w,x,y,z\} \rightarrow \{0,1,2,3,4,5,6,7,8,9\}$$ be $$f = \{(s,4),(t,8),(u,8),(v,1),(w,2),(x,4),(y,6),(z,4)\}$$.

This f is neither injective nor surjective, so it certainly is not invertible. Be sure you understand the following statements.

1. $$f(\{s,t,u,z\}) = \{8,4\}$$

2. $$f(\{s,x,z\}) = \{4\}$$

3. $$f(\{s,v,w,y\}) = \{1,2,4,6\}$$

4. $$f(\emptyset) = \emptyset$$

5. $$f^{-1}(\{4\}) = \{s,x,z\}$$

6. $$f^{-1}(\{4,9\})= \{s,x,z\}$$

7. $$f^{-1}(\{9\}) = \emptyset$$

8. $$f^{-1}(\{1,4,8\}) = \{s,t,u,v,x,z\}$$

It is important to realize that the X and Y in Definition 12.9 are subsets (not elements!) of A and B. In Example 12.13 we had $$f^{-1}(\{4\}) = \{s, x, z\}$$, while $$f^{-1}(4)$$ is meaningless because the inverse function $$f^{-1}$$ does not exist. And there is a subtle difference between $$f(\{s\}) = \{4\}$$ and $$f(s) = 4$$. Be careful.

Example 12.14

Consider the function $$f : \mathbb{R} \rightarrow \mathbb{R}$$ defined as $$f(x) = x^2$$. Note that $$f(\{0,1,2\}) = \{0,1,4\}$$ and $$f^{-1}(\{0,1,4\}) = \{-2,-1,0,1,2\}$$. This shows that $$f^{-1}(f(X)) \ne X$$ in general.

Using the same f, check your understanding of these statements about images and preimages of intervals: $$f ([-2,3]) = [0,9]$$, and $$f^{-1}([0,9]) = [-3,3]$$. Also $$f(\mathbb{R}) = [0,\infty)$$ and $$f^{-1}([-2,-1]) = \emptyset$$.

If you continue with mathematics you will likely encounter the following results. For now, you are asked to prove them in the exercises.

Theorem 12.4

Given $$f :A \rightarrow B$$, let $$W, X \subseteq A$$, and $$Y, Z \subseteq B$$. Then

1. $$f(W \cap X) \subseteq f(W) \cap f(X)$$

2. $$f(W \cup X) = f(W) \cup f(X)$$

3. $$X \subseteq f^{-1}f(X)$$

4. $$f^{-1}(Y \cup Z) = f^{-1}(Y) \cup f^{-1}(Z)$$

5. $$f^{-1}(Y \cap Z) = f^{-1}(Y) \cap f^{-1}(Z)$$

6. $$f(f^{-1}(Y)) \subseteq Y$$.

## Exercise

Exercise $$\PageIndex{1}$$

Consider the function $$f : \mathbb{R} \rightarrow \mathbb{R}$$ defined as $$f(x) = x^2+3$$. Find $$f([-3,5])$$ and $$f^{-1}([12, 19])$$.

Exercise $$\PageIndex{2}$$

Consider the function $$f : \{1,2,3,4,5,6,7\} \rightarrow \{0,1,2,3,4,5,6,7,8,9\}$$ given as $$f = \{(1,3), (2,8), (3,3), (4,1), (5,2), (6,4), (7,6)\}$$.

Find: $$f(\{1,2,3\} , f(\{4,5,6,7\}), f (\emptyset), f(\{0,5,9\})$$ and $$f(\{0,3,5,9\})$$.

Exercise $$\PageIndex{3}$$

This problem concerns functions $$f : \{1,2,3,4,5,6,7\} \rightarrow \{0,1,2,3,4\}$$. How many such functions have the property that $$|f^{-1}(\{3\})| = 3$$?

Exercise $$\PageIndex{4}$$

This problem concerns functions $$f : \{1,2,3,4,5,6,7,8\} \rightarrow \{0,1,2,3,4,5,6\}$$. How many such functions have the property that $$|f^{-1}(\{2\})| = 4$$?

Exercise $$\PageIndex{5}$$

Consider a function $$f : A \rightarrow B$$ and a subset $$X \subseteq A$$. We observed in Example 12.14 that $$f^{-1}(f(X)) \ne X$$ in general. However $$X \subseteq f^{-1}(f(X))$$ is always true. Prove this.

Exercise $$\PageIndex{6}$$

Given a function $$f : A \rightarrow B$$ and a subset $$Y \subseteq B$$, is $$f(f^{-1}(Y)) = Y$$ always true? Prove or give a counterexample.

Exercise $$\PageIndex{7}$$

Given a function $$f : A \rightarrow B$$ and subsets $$W,X \subseteq A$$, prove $$f(W \cap X) \subseteq f(W) \cap f(X)$$.

Exercise $$\PageIndex{8}$$

Given a function $$f : A \rightarrow B$$ and subsets $$W,X \subseteq A$$, then $$f(W \cap X) = f(W) \cap f(X)$$ is false in general. Produce a counterexample.

Exercise $$\PageIndex{9}$$

Given a function $$f : A \rightarrow B$$ and subsets $$W,X \subseteq A$$, prove $$f(W \cup X) = f(W) \cup f(X)$$.

Exercise $$\PageIndex{10}$$

Given $$f : A \rightarrow B$$ and subsets $$Y,Z \subseteq B$$, prove $$f^{-1}(Y \cap Z) = f^{-1}(Y) \cap f^{-1}(Z)$$.

Exercise $$\PageIndex{11}$$

Given $$f : A \rightarrow B$$ and subsets $$Y,Z \subseteq B$$, prove $$f^{-1}(Y \cup Z) = f^{-1}(Y) \cup f^{-1}(Z)$$.

Exercise $$\PageIndex{12}$$

Consider $$f : A \rightarrow B$$. Prove that f is injective if and only if $$X = f^{-1}(f(X))$$ for all $$X \subseteq A$$. Prove that f is surjective if and only if $$f(f^{-1}(Y)) =Y$$ for all $$Y \subseteq B$$.

Exercise $$\PageIndex{13}$$

Let $$f : A \rightarrow B$$ be a function, and $$X \subseteq A$$. Prove or disprove: $$f(f^{-1}(f(X))) = f(X)$$.

Exercise $$\PageIndex{14}$$

Let $$f : A \rightarrow B$$ be a function, and $$Y \subseteq B$$. Prove or disprove: $$f^{-1}(f(f^{-1}(f(X))) = f^{-1}(Y)$$.