Determine whether each of the functions below is onto andor

Determine whether each of the functions below is onto, and/or one-to-one for f: Z -> Z, prove your answer.

8 points Determine whether each of the functions below is onto, and/or one-to-one for f z Z, prove your answers

Solution

The function is f(x) = 2 x^3.

^{Find one-to-one :}

if f(x)=f(x\') iff x=x\'

2 (x)^3. = 2 (x\')^3.

(x)^3. = (x\')^3.

^{Apply cube root on each side.}

x^3. = x\'

Therefore it is one-to-one function.

Find onto function :

let f(x) = y

substitute f(x) = 2x³  in the above.

2x³ = y

sove for x.

x³ = y/2

x = (y/2)^(1/3)

f(y) = 2 ( (y/2)^(1/3) )³

f(y) = 2 ( (y/2)

f(y) = y

Therefore it is onto function.