For each of the following functions determine if the functio

For each of the following functions, determine if the function is \' and determine if the function is a surjection. Justify all conclusions. f:Z rightarrow Z defined by f(x) = 3x + 1, for all x epsilon Z. F: Q rightarrow Q defined by F(x) = 3x + 1, for all x epsilon Q g:R rightarrow R defined by g(x) = x^3, for all x epsilon R. G:Q rightarrow Q defined by G(x) = x^3, for all x epsilon Q. k:R rightarrow R defined by k(x) = -x^2, for all x epsilon R. K:R* rightarrow R defined by K(x) = e^-x^2, for all x epsilon R. K_1:R* rightarrow T defined by K_1(x) = e^-x^2, for all x epsilon R*, when T = {y epsilon R| 0

Solution

Given function g: R ...R is defined as g(x) = x^3 . let g be a function whose domain is R , the function g is injective iff for all x and y in R , if g(x) = g(y), then x = y;

g(x) = g(y)

implies: x^3 = y^3

implies: x = y

g(x) = x^3 is injective .

if the function g(x ) is surjective where every element in the codomain is a valied out put of the function . that is Range = codomain.

g(x) = x^3 is also surjective because the range of all the numbers from [y^(1/3)]^3 = y.

there foure g(x) = x^3 is surjective.