In exercises 1-5, let the table below define a mapping of a set into a set . Set contains points C, D, E, F and G, and set contains points H, I, J, K, L and M.
Pairing:
C H,
D I,
E J,
F K,
G L,
M.

1. Is mapped into or onto set ?

Ans:Into. The point M is not in the image of the mapping.

2.What is the domain of the mapping?

Ans: {C, D, E, F, G}

3. What is the codomain of the mapping?

Ans: {H, I, J, K, L, M}

4. What is the range of the mapping?

Ans: {H, I, J, K, L}

5. Is the mapping one-to-one?

Ans: Yes.

7. Which of the mappings shown below are transformations?

Solution:Only (c) is a transformation. (a) is not one-to-one and (b) is not onto.

9. What is the image of -2 under the transformation indicated by x 2x - 1 ?

Ans: -5.

11. What is the image of the point (2,1) under the transformation indicated by (x,y) (x + 5, y - 3) ?

Ans:(7, -2).

13. What is the image of the point (-3, -2) under the transformation indicated by (x,y) (x + 5, y - 3) ?

Ans:(2, -5).

For exercises 14-17, let f = {(a,b), (c,d), (e,h)} and g = {(b,i), (d,j), (h,k)} be transformations of points on a line.

15. Find the product fg.

Solution: This is undefined since the range of g does not lie in the domain of f.

17. Find g^-1.

Ans: g^-1 = {(i,b), (j,d), (k,h)}.

For exercises 18-21, let f be the transformation such that (x,y) has the image (x-5, y+2) and g be the transformation such that (x,y) has the image (x+2, y-3).

19. Find the product gf.

Ans: (x,y) (x-3, y-1).

21. Find g^-1.

Ans: (x,y) (x-2, y+3).

23. Find the screen coordinates for the point with world coordinates (32,18) if l = 25, l' = 30, h = 20, h' = 18, x = y = 5, x' = y' = 20.

Ans: (52.4, 31.7) Solution: m' = (30/25) (32 - 5) + 20 and n' = (18/20)(18 - 5) + 20.

24. If the product T₂T₁ exists, then the transformation T₂ is said to be compatible with T₁. If T₂ is compatible with T₁, is T₁ necessarily compatible with T₂?

Ans: No. Solution: The existence of the composition of two functions in one order does not imply the existence of the composition in the other order.

26. A transformation f such that ff = I, the identity, is called an involutory transformation. If f is an involutory transformation, prove that f = f^-1.

Solution: By definition, the inverse of f (i.e., f^-1) is a transformation g such that fg = gf = I. Since f is involutory, ff = I and so, f can itself play the role of g in the definition, making f = f^-1.