Quiz:Piecewise definition of function

This quiz is related to piecewise definition of function.

Converting to and from piecewise definitions

1 Which of the following is the correct piecewise linear definition for $|x+1|-\left|x\right|$?

	$\{\begin{array}{cc}1,& x\ge -1\\ -1,& x<-1\\ \end{array}$
	$\{\begin{array}{cc}1,& x\ge 0\\ 0,& -1<x<0\\ -1,& x\le -1\\ \end{array}$
	$\{\begin{array}{cc}1,& x\ge 0\\ 2x+1,& -1<x<0\\ -1,& x\le -1\\ \end{array}$
	$\{\begin{array}{cc}2x+1,& x\ge -1/2\\ -2x-1,& x<-1/2\\ \end{array}$
	$\{\begin{array}{cc}1,& x\ge 0\\ 2x+1,& -1/2<x<0\\ -2x-1,& -1\le x\le -1/2\\ -1,& x\le -1\\ \end{array}$

2 Suppose $a<b$ are real numbers. For an real number $x$, define $f(x)$ as the minimum of the distances from $x$ to $a$ and $b$. In other words, $f(x):=min\{|x-a|,|x-b|\}$. Which of the following is the correct piecewise linear definition of $f$?

	$\{\begin{array}{cc}a-x,& x\le a\\ x-(a+b)/2,& a<x<b\\ x-b,& x\ge b\\ \end{array}$
	$\{\begin{array}{cc}a-x,& x\le a\\ x-a,& a<x<b\\ x-b,& x\ge b\\ \end{array}$
	$\{\begin{array}{cc}a-x,& x\le a\\ b-x,& a<x<b\\ x-b,& x\ge b\\ \end{array}$
	$\{\begin{array}{cc}a-x,& x\le a\\ x-a,& a<x<(b-a)/2\\ b-x,& (b-a)/2\le x<b\\ x-b,& x\ge b\\ \end{array}$
	$\{\begin{array}{cc}a-x,& x\le a\\ x-a,& a<x<(b+a)/2\\ b-x,& (b+a)/2\le x<b\\ x-b,& x\ge b\\ \end{array}$

Pointwise combination (computational)

Suppose $f(x):=\{\begin{array}{cc}{x}^2,& x<2\\ x^3,& x\ge 2\\ \end{array}$ and $g(x):=\{\begin{array}{cc}x+1,& x\le 1\\ 2x+3,& x>1\\ \end{array}$. What is $(f+g)(x)$?

	$\{\begin{array}{cc}{x}^2+x+1,& x\le 3\\ 3x+3,& x>3\\ \end{array}$
	$\{\begin{array}{cc}{x}^2+x+1,& x<3\\ 3x+3,& x\ge 3\\ \end{array}$
	$\{\begin{array}{cc}{x}^2+x+1,& x\le 1\\ x^2+2x+3,& 1<x<2\\ x^3+2x+3,& x\ge 2\\ \end{array}$
	$\{\begin{array}{cc}{x}^2+x+1,& x<1\\ x^2+2x+3,& 1\le x\le 2\\ x^3+2x+3,& x>2\\ \end{array}$
	$\{\begin{array}{cc}{x}^2+x+1,& x<2\\ x^3+2x+3,& 2\le x\le 3\\ x^2+2x+3,& x>3\\ \end{array}$

Continuity and pointwise combination

1 Suppose $f$ and $g$ are functions defined on a closed interval $[a,b]$ and are both piecewise continuous, i.e., each function is continuous except possibly at finitely many points in the open interval $(a,b)$. Then, which of the following functions is not guaranteed to be a piecewise continuous function on $[a,b]$?

	$f+g$, the pointwise sum of functions
	$f-g$, the pointwise difference of functions
	$f\cdot g$, the pointwise product of functions
	None of the above, i.e., they are all guaranteed to be piecewise continuous functions on $[a,b]$
	All of the above, i.e., none of them is guaranteed to be a piecewise continuous function on $[a,b]$

2 Suppose $f$ and $g$ are functions defined on all of $\mathbb{R}$. $f$ is discontinuous at 5 points and $g$ is discontinuous at 3 points. What can we say about $f+g$?

	It is discontinuous at exactly 8 points
	It is discontinuous at at least 8 points
	It is discontinuous at at most 2 points
	It is discontinuous at at least 2 points and at most 8 points
	It may be discontinuous at an arbitrarily large number of points.

3 Suppose $f$ and $g$ are functions defined on all of $\mathbb{R}$. $f$ is discontinuous at 5 points and $g$ is discontinuous at 3 points. What can we say about $f\cdot g$?

	It is discontinuous at exactly 8 points
	It is discontinuous at at least 8 points
	It is discontinuous at at most 8 points
	It is discontinuous at at least 2 points
	It is discontinuous at at most 2 points

4 Suppose $f$ and $g$ are functions defined on all of $\mathbb{R}$. Suppose $f$ is continuous and piecewise linear, with different nonconstant linear piece definitions on the interval $(-\mathrm{\infty },0],[0,2],[2,\mathrm{\infty })$. Suppose $g$ is continuous and piecewise linear with different piece definitions on $(-\mathrm{\infty },1],[1,3],[3,\mathrm{\infty })$. What can we say about the pointwise product of functions $f\cdot g$?

	It is continuous and piecewise linear, with (potentially) different piece definitions on the intervals $(-\mathrm{\infty },0],[0,1],[1,2],[2,3],[3,\mathrm{\infty })$
	It is continuous and piecewise quadratic, with (potentially) different piece definitions on the intervals $(-\mathrm{\infty },0],[0,1],[1,2],[2,3],[3,\mathrm{\infty })$
	It is continuous and linear with a single piece definition
	It is continuous and quadratic with a single piece definition
	It is linear but need not be continuous

Composition

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Differentiation

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