Homework 8

Details

1. (6) Let \(f:\mathbb{Z}\rightarrow\mathbb{Z}\) be defined by \(f(x)=x^3-x\). Answer each of the following, with a brief justification.
   1. Is \(f\) one-to-one?
   2. Is \(f\) onto?
   3. Is \(f\) invertible?
2. (8) Let \(f\) be a function that maps a person on earth to the number of pets that they have owned.
   1. What is the domain of \(f\)?
   2. What is the codomain of \(f\)?
   3. The exact range of \(f\) is difficult to know for sure. But is it the same as the codomain? Explain.
   4. Is the range of \(f\) a subset of the codomain? Explain.
3. (6) Let \(f(x)=e^x\) and \(g(x)=3x^2-4x+7\)
1. Compute \((f\circ g)(x)\)
2. Compute \((g\circ f)(x)\)