Show that the function f: N → Z, defined by\(f(x)=\begin{cases}\frac

1.	Show that the function f: N → Z, defined by\(f(x)=\begin{cases}\frac{1}{2}(n-1)\text{ when n is odd}\\-\frac{1}{2}n,\text{ when n is even}\end{cases}\)is both one - one and onto.
Answer» \(f(x)=\begin{cases}\frac{1}{2}(n-1)\text{ when n is odd}\\-\frac{1}{2}n,\text{ when n is even}\end{cases}\) f(1) = 0 f(2) = - 1 f(3) = 1 f(4) = - 2 f(5) = 2 f(6) = - 3 Since at no different values of x we get same value of y ∴f(n) is one –one And range of f(n) = Z = Z(codomain) ∴ the function f: N → Z, defined by \(f(x)=\begin{cases}\frac{1}{2}(n-1)\text{ when n is odd}\\-\frac{1}{2}n,\text{ when n is even}\end{cases}\) is both one - one and onto.

Show that the function f: N → Z, defined by\(f(x)=\begin{cases}\frac{1}{2}(n-1)\text{ when n is odd}\\-\frac{1}{2}n,\text{ when n is even}\end{cases}\)is both one - one and onto.

Answer»

\(f(x)=\begin{cases}\frac{1}{2}(n-1)\text{ when n is odd}\\-\frac{1}{2}n,\text{ when n is even}\end{cases}\)

f(1) = 0

f(2) = - 1

f(3) = 1

f(4) = - 2

f(5) = 2

f(6) = - 3

Since at no different values of x we get same value of y ∴f(n) is one –one

And range of f(n) = Z = Z(codomain)

∴ the function f: N → Z, defined by

\(f(x)=\begin{cases}\frac{1}{2}(n-1)\text{ when n is odd}\\-\frac{1}{2}n,\text{ when n is even}\end{cases}\)

is both one - one and onto.