[Not to be turned in]
- Suppose $x$ is a real number with $x>-1$. Prove by induction that $(1+x)^n \ge 1+nx$ for all natural numbers $n$.
(Note that you will have to use the fact that $x>-1$ in some way, so make sure that is part of your proof.)
- Suppse $A = \{1,2\}$, $B = \{a,b,c\}$ and $C = \{3,4,5\}$. Give an example of a function $f\colon A\times B\to C$. Give your function in both the rule-based and graph-based forms.
(Hint: for the rule-based form, you probably will not be able to use a formula. Instead, since the sets are finite, you can do this by explicitly listing the value of $f$ for each input.)
- If $f\colon \R\to\R$ is defined by
$\displaystyle f(x)=\cases{ 3-2x&\text{if $x\le 1$}\cr x(x-2)&\text{if $x\ge 1$} } $
which of the properties of the definition of a function does $f$ satisfy and which does it fail? Justify your answers. Is $f$ a function?
|
|