Question

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

Answer 1

a. Since
`f:\Bbb Z\to\Bbb z`, the function
`f(x)=2x^2+1` is not onto because not every element in the codomain
`\Bbb Z` is represented by
`f(x)`. For any
`x\in Z`, we have
`x^2\ge0`, so
`x^2+1\ge1`, which means there is no choice of
`x` such that
`f(x)\le0`.

b.
`f` is not one-to-one because there are infinitely many choices of
`x` for which
`f(-x)=f(x)`. For example,
`f(1)=1^2+1=2` and
`f(-1)=(-1)^2+1=2`.