1 Consider the function f R2 R2 defined by fa b 2a b 2b a a

1. Consider the function f: R2 R2 defined by f(a, b) = 2a b, 2b a

a. Calculate f(0,0), f(1,0), f(0,1), f(1,1), f(2, 3)

b. What is the image of the line x = 1 under f ?

c. What is the image of the line y = 1 under f ?

d. What is the image of the line y = x under f ?

Solution

f(a, b) = 2a b, 2b a

a) f(0,0):

Plugging a=0 and b=0

f(0, 0) = 2(0) (0), 2(0) (0) = 0, 0

f(1,0): Plugging a=1 and b=0

f(1, 0) = 2(1) (0), 2(0) (1) = 2, -1

f(1,1): Plugging a=1 and b=1

f(1, 1) = 2(1) (1), 2(1) (1) = 1, 1

f(2,-3): Plugging a=2 and b=-3

f(2, -3) = 2(2) (-3), 2(-3) (2) = 7, -8

b) For image of line x=1 under f, we need to plug a=1 and keep b same.

f(a, b) = 2a b, 2b a => f(1, b) = 2(1) b, 2b (1) = 2 b, 2b 1

c) For image of line y=1 under f, we need to plug b=1 and keep a same.

f(a, b) = 2a b, 2b a => f(a, 1) = 2a 1, 2(1) a = 2a 1, 2 a

d) For image of line y=x under f, we need to plug b=a and keep a same.

f(a, b) = 2a b, 2b a => f(a, a) = 2a a, 2(a) a = a, a