Let f={(2,3),(4,9),(3,-8)} and g={(1,2),(2,5),(3,10),(4,17)}. Determine:
a)-2f b)fg c)fg(2) d)g2
I can't see all of the answer
fg(2) = f(2)*g(2) = 3*5 = 15
if by g2 you mean g^2, then that would be {(1,4),(2,25),(3,100),(4,289)}
a) -2f: Well, if we apply a little arithmetic to each point in f, we get:
-2(2,3) = (-4,-6)
-2(4,9) = (-8,-18)
-2(3,-8) = (-6,16)
So, -2f = {(-4,-6), (-8,-18), (-6,16)}
b) fg: To find fg, we need to combine each point of g with each point of f. So we get:
(1,2)(2,3) = (1*2, 2+3) = (2,5)
(1,2)(4,9) = (1*4, 2+9) = (4,11)
(1,2)(3,-8) = (1*3, 2-8) = (3,-6)
(2,5)(2,3) = (2*2, 5+3) = (4,8)
(2,5)(4,9) = (2*4, 5+9) = (8,14)
(2,5)(3,-8) = (2*3, 5-8) = (6,-3)
(3,10)(2,3) = (3*2, 10+3) = (6,13)
(3,10)(4,9) = (3*4, 10+9) = (12,19)
(3,10)(3,-8) = (3*3, 10-8) = (9,2)
(4,17)(2,3) = (4*2, 17+3) = (8,20)
(4,17)(4,9) = (4*4, 17+9) = (16,26)
(4,17)(3,-8) = (4*3, 17-8) = (12,9)
So, fg = {(2,5), (4,11), (3,-6), (4,8), (8,14), (6,-3), (6,13), (12,19), (9,2), (8,20), (16,26), (12,9)}
c) fg(2): To find fg(2), we need to substitute x = 2 in each point of fg:
(2,5)(2) = (2*2, 5*2) = (4,10)
(4,11)(2) = (4*2, 11*2) = (8,22)
(3,-6)(2) = (3*2, -6*2) = (6,-12)
(4,8)(2) = (4*2, 8*2) = (8,16)
(8,14)(2) = (8*2, 14*2) = (16,28)
(6,-3)(2) = (6*2, -3*2) = (12,-6)
(6,13)(2) = (6*2, 13*2) = (12,26)
(12,19)(2) = (12*2, 19*2) = (24,38)
(9,2)(2) = (9*2, 2*2) = (18,4)
(8,20)(2) = (8*2, 20*2) = (16,40)
(16,26)(2) = (16*2, 26*2) = (32,52)
(12,9)(2) = (12*2, 9*2) = (24,18)
So, fg(2) = {(4,10), (8,22), (6,-12), (8,16), (16,28), (12,-6), (12,26), (24,38), (18,4), (16,40), (32,52), (24,18)}
d) g^2: To find g^2, we need to apply the g function on g. So we get:
(1,2)(1,2) = (1*1, 2+2) = (1,4)
(1,2)(2,5) = (1*2, 2+5) = (2,7)
(1,2)(3,10) = (1*3, 2+10) = (3,12)
(1,2)(4,17) = (1*4, 2+17) = (4,19)
(2,5)(1,2) = (2*1, 5+2) = (2,7)
(2,5)(2,5) = (2*2, 5+5) = (4,10)
(2,5)(3,10) = (2*3, 5+10) = (6,15)
(2,5)(4,17) = (2*4, 5+17) = (8,22)
(3,10)(1,2) = (3*1, 10+2) = (3,12)
(3,10)(2,5) = (3*2, 10+5) = (6,15)
(3,10)(3,10) = (3*3, 10+10) = (9,20)
(3,10)(4,17) = (3*4, 10+17) = (12,27)
(4,17)(1,2) = (4*1, 17+2) = (4,19)
(4,17)(2,5) = (4*2, 17+5) = (8,22)
(4,17)(3,10) = (4*3, 17+10) = (12,27)
(4,17)(4,17) = (4*4, 17+17) = (16,34)
So, g^2 = {(1,4), (2,7), (3,12), (4,19), (2,7), (4,10), (6,15), (8,22), (3,12), (6,15), (9,20), (12,27), (4,19), (8,22), (12,27), (16,34)}
a) To find -2f, we multiply each coordinate in f by -2.
-2f = {(-4,-6), (-8,-18), (-6,16)}
b) To find the product fg, we need to match up x-values from f and g and multiply the corresponding y-values.
fg = {(2,3)*(2,5), (4,9)*(4,17), (3,-8)*(3,10)}
= {(4,15), (16,153), (9,-80)}
c) To find fg(2), we need to substitute x = 2 into the function fg.
fg(2) = {(2,3)*(2,5), (4,9)*(4,17), (3,-8)*(3,10)}(2)
= {(4,15), (16,153), (9,-80)}(2)
= (16,153)
d) To find g^2, we need to perform the operation g(g).
g^2 = g(g) = g{(1,2)*(1,2), (2,5)*(2,5), (3,10)*(3,10), (4,17)*(4,17)}
= g{(1,4), (4,25), (9,100), (16,289)}
= {(1,4)*(1,4), (4,25)*(4,25), (9,100)*(9,100), (16,289)*(16,289)}
= {(1,16), (16,625), (81,10000), (256,83521)}
= {(1,16), (16,625), (81,10000), (256,83521)}
To determine the required operations, we need to understand some basic operations on sets and functions:
1. Scalar multiplication: Multiplying a set of points by a scalar value involves multiplying each point's y-coordinate by the scalar value.
2. Function composition: Composing two functions involves applying one function after the other. For example, if f(x) = 2x and g(x) = x + 3, then (f∘g)(x) = f(g(x)) = f(x + 3) = 2(x + 3) = 2x + 6.
Now let's solve the given questions step by step:
a) -2f:
To obtain -2f, we need to multiply each y-coordinate of the points in set f by -2.
-2f = {(-2*2, -2*3), (-2*4, -2*9), (-2*3, -2*(-8))} = {(-4, -6), (-8, -18), (-6, 16)}.
b) fg:
To find the product of two sets of points, we need to find all possible points generated by combining each point in the first set with each point in the second set.
fg = {(1+2, 2+3), (2+2, 5+3), (3+2, 10+3), (4+2, 17+3)} = {(3, 5), (4, 8), (5, 13), (6, 20)}.
c) fg(2):
To find fg(2), we need to apply the composition function fg with the input 2.
fg(2) = (f∘g)(2) = f(g(2)).
First, evaluate g(2) by finding the point in set g with x=2: g(2) = (2, 5).
Then, apply f to g(2): f(g(2)) = f(2, 5) = (2*2, 2*5) = (4, 10).
Therefore, fg(2) = (4, 10).
d) g^2:
To calculate g^2, we need to compose g with itself (g∘g).
First, calculate g(x) and then substitute it into g(x):
g(x) = {(1+3, 2+3), (2+3, 5+3), (3+3, 10+3), (4+3, 17+3)} = {(4, 5), (5, 8), (6, 13), (7, 20)}.
Now apply g to g(x):
g^2 = {(4+3, 5+3), (5+3, 8+3), (6+3, 13+3), (7+3, 20+3)} = {(7, 8), (8, 11), (9, 16), (10, 23)}.
Therefore, the solutions are:
a) -2f = {(-4, -6), (-8, -18), (-6, 16)}
b) fg = {(3, 5), (4, 8), (5, 13), (6, 20)}
c) fg(2) = (4, 10)
d) g^2 = {(7, 8), (8, 11), (9, 16), (10, 23)}