Determine which of the following functions are onetoone f R2

Determine which of the following functions are one-to-one f: R^2 rightarrow R^2 defined by f (x, y) = (2 + ym 2x + 2y) R^3 rightarrow R^3 defined by f (x, y, z) = (x - y, y - z, x - z) R^2 rightarrow R^2 defined by f (x, y) = (x + y, x - y) R rightarrow R defined by f (x) = x^3. R rightarrow R defined by f (x) = x^3 - x.

Solution

A.

Not one to one

Example

f(x,y)=f(x+1,y-1)

B.

Not one to one

f(x,y,z)=f(x+a,y+a,z+a) for any real number a

C.

one to one

Let, f(x,y)=f(x\',y\')

x+y=x\'+y\'

x-y=x\'-y

Solvling gives, x=x\',y=y\'

D.

One to one

f(x)=f(y)

x^3=y^3

(x-y)(x^2+y^2+xy)=0

So, x=y

E.

Not one to one

Because, f(0)=f(1)=0