12.5.2) f : R-{2} → R-{5} defined as f(x) = (5x+1)/(x-2) is bijective. Find its inverse.

y = (5x+1)/(x-2) y = 5 + 9/(x-2) y-5 = 9/(x-2) x-2 = 9/(y-5) x = 9/(y-5) + 2

f^-1(y) = 9/(y-5) + 2

12.5.4) The function f : R → (0,∞) defined as f (x) = $e^{x^3+1}$ is bijective. Find its inverse.

y = $e^{x^3+1}$ ln(y) = x³ + 1 cube-root(ln(y) - 1) = x f^-1(y) = cube-root(ln(y) - 1)

12.5.6) The function f :Z×Z→Z×Z defined by the formula f(m,n)= (5m+4n,4m+3n) is bijective. Find its inverse.

(x,y) = (5m+4n,4m+3n)

x = 5m+4n y = 4m+3n

4x = 20m+16n 5y = 20m + 15n

4x-5y = n

m = n = 4x-5y

f^-1(x,y) = (..., 4x-5y)

12.6.2) Consider the function f : {􏰛1,2,3,4,5,6,7}􏰜 → {􏰛0,1,2,3,4,5,6,7,8,9}􏰜 given as f = {(1,3), (2,8), (3,3), (4,1), (5,2), (6,4), (7,6)}. Find: f({1,2,3}) , f({4,5,6,7}) , f ({ }􏰀), f^-1({0,5,9} and f^-1({0,3,5,9}).

f({1,2,3}) = {3,8} f({4,5,6,7}) = {1,2,4,6} f ({ }􏰀) = { } f^-1({0,5,9} = { } f^-1({0,3,5,9}) = {1,3}