For the following functions f g belonging to FZ compute fg a

For the following functions f, g belonging to F(Z), compute fg and gf.

(b) f(n)= { n/2, if n is even and g(n)= { n-1 , if n is even

{ n+1 if n is odd { 2n , if n is odd

Determine which of the functions are invertible. If a function f is invertible, find f^-1.

(b) f: {-1,0,1} —> {pi, e, sqrt2} defined by f(-1) = e, f(0) = pi, and f(1) = sqrt2

(d) f: R —> R defined by f(x) = 7x+3

(k) f: Z —> Z x {0,1} defined by f(n) = { (n/2, 0) , if n belongs to E

(n-1/2, 1) if n belongs to O

b) Given that f(n)= { n/2, if n is even and g(n)= { n-1 , if n is even

{ n+1 if n is odd { 2n , if n is odd

n = even

fg(n) = f [g(n)] = f [n-1] = n-1/2

gf(n) = g [f(n)] = g [n/2] = (n/2)-1 = n-2/2

n = odd

fg(n) = f [g(n)] = f [2n] = 2n+1

gf(n) = g [f(n)] = g [n+1] = 2(n+1) = 2n+2

b) f: {-1,0,1} —> {pi, e, sqrt2} defined by f(-1) = e, f(0) = pi, and f(1) = sqrt2

f^-1(e) = -1

f^-1(pi) = 0

f-1(sqrt2) = 1

c) f: Z —> Z defined by f(x) = 7x+3

To find the inverse, interchange x and y.

x = 7y + 3

Now isolate for y!!

7y = x-3

y= (x-3)/7

Therefore,

f^-1(x) = (x-3)/7

d) f: R —> R defined by f(x) = 7x+3

o find the inverse, interchange x and y.