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 $f(-1)=(-1)^2+1=2$.