**Functions**

- Let
*A* = {*q*, *r*, *s*,* t*} and *B* = {17, 18, 19, 20}. Determine which of the following are functions. Explain why or why not.

2. Give the ranges of each of these functions where sets A and B are as stated in #1:

- }

3. State whether each of these functions, where sets A and B are as stated in #1, are one-to-one, onto, both, or neither, and give a brief explanation for each answer:

- }

4. Determine all bijections from *A* into *B*.

a. *A* = {*q*,* r*, *s*} and *B* = {2, 3, 4}

b. *A* = {1, 2, 3, 4} and *B* = {5, 6, 7, 8}

5. Which of the following functions fromare one-to-one, onto, or both? Prove your answers.

a.

b.

c.

d.

e.

6. For each function in parts a through f, state a domain that, if it was the domain of the given function, would make the function one-to-one, and explain your answer. If no such domain exists, explain why not. (Hint: graph the function and use the appropriate line test).

a.

b.

c.

d.

e.

f. |

7. For each function in parts a through f, state a codomain that, if it was the codomain of the given function, would make the function onto, and explain your answer. If no such codomain exists, explain why not.

a.

b.

c.

d.

e.

f. |

8. Let *f*= {(-2, 3), (-1, 1), (0, 0), (1, -1), (2, -3)} and

let *g* = {(-3, 1), (-1, -2), (0, 2), (2, 2), (3, 1)}. Find:

a. *f*(1)

b. *g*(-1)

c.

d.

e.

9. Define *q*, *r*, and *s*, all functions on the integers, by,, and. Determine:

a.

b.

c.

d.

10. Consider the functions *f*, *g* (both on the reals) defined byand.

a. Show that *f*is injective.

b. Show that *f* is surjective.

c. Find.

d. Find.

e. Find.