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Homework on 28 October 2019:

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  1. 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.)

  2. 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.)

  3. 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?


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Created: 28 Oct 2019
Last modified: 28 Oct 2019 at 4:19 PM
Comments to: dpvc@union.edu
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