Suppose f Z rightarrow Z is defined by fx 2x2 1 Explain us

Suppose f: Z rightarrow Z is defined by f(x) = 2x^2 + 1. Explain using the definition of onto functions why the function f is or is not an onto function. Explain using the definition of one-to-one functions why the function f is or is not a 1-1 function.

Solution

a.

For any integer x ,2x^2+1 is an odd integer

So range of f consists only of odd integers and no even integesr

Hence, f is not onto and codomain of f is Z

b.

f(1)=2*1+1=3

f(-1)=2*(-1)^2+1=3

Hence, f is not one to one