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 $|x+1|-|x|$?

	$\left\lbrace {\begin{array}{rl}1,&x\geq -1\\-1,&x<-1\\\end{array}}\right.$
	$\left\lbrace {\begin{array}{rl}1,&x\geq 0\\0,&-1<x<0\\-1,&x\leq -1\\\end{array}}\right.$
	$\left\lbrace {\begin{array}{rl}1,&x\geq 0\\2x+1,&-1<x<0\\-1,&x\leq -1\\\end{array}}\right.$
	$\left\lbrace {\begin{array}{rl}2x+1,&x\geq -1/2\\-2x-1,&x<-1/2\\\end{array}}\right.$
	$\left\lbrace {\begin{array}{rl}1,&x\geq 0\\2x+1,&-1/2<x<0\\-2x-1,&-1\leq x\leq -1/2\\-1,&x\leq -1\\\end{array}}\right.$

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 $f$?

	$\left\lbrace {\begin{array}{rl}a-x,&x\leq a\\x-(a+b)/2,&a<x<b\\x-b,&x\geq b\\\end{array}}\right.$
	$\left\lbrace {\begin{array}{rl}a-x,&x\leq a\\x-a,&a<x<b\\x-b,&x\geq b\\\end{array}}\right.$
	$\left\lbrace {\begin{array}{rl}a-x,&x\leq a\\b-x,&a<x<b\\x-b,&x\geq b\\\end{array}}\right.$
	$\left\lbrace {\begin{array}{rl}a-x,&x\leq a\\x-a,&a<x<(b-a)/2\\b-x,&(b-a)/2\leq x<b\\x-b,&x\geq b\\\end{array}}\right.$
	$\left\lbrace {\begin{array}{rl}a-x,&x\leq a\\x-a,&a<x<(b+a)/2\\b-x,&(b+a)/2\leq x<b\\x-b,&x\geq b\\\end{array}}\right.$

3 Suppose $a_{1}<a_{2}<\dots <a_{n}$ are $n$ distinct real numbers. For an real number $x$, define $f(x)$ as the minimum of the distances from $x$ to each of the numbers $a_{1},a_{2},\dots ,a_{n}$. In other words, $\!f(x):=\min\{|x-a_{1}|,|x-a_{2}|,|x-a_{3}|,\dots ,|x-a_{n}|\}$. How many different linear pieces does the correct piecewise linear definition of $f$ have?

	$n$
	$n+1$
	$2n-2$
	$2n-1$
	$2n$

Pointwise combination (computational)

Suppose $f(x):=\left\lbrace {\begin{array}{rl}x^{2},&x<2\\x^{3},&x\geq 2\\\end{array}}\right.$ and $g(x):=\left\lbrace {\begin{array}{rl}x+1,&x\leq 1\\2x+3,&x>1\\\end{array}}\right.$. What is $(f+g)(x)$?

	$\left\lbrace {\begin{array}{rl}x^{2}+x+1,&x\leq 3\\3x+3,&x>3\\\end{array}}\right.$
	$\left\lbrace {\begin{array}{rl}x^{2}+x+1,&x<3\\3x+3,&x\geq 3\\\end{array}}\right.$
	$\left\lbrace {\begin{array}{rl}x^{2}+x+1,&x\leq 1\\x^{2}+2x+3,&1<x<2\\x^{3}+2x+3,&x\geq 2\\\end{array}}\right.$
	$\left\lbrace {\begin{array}{rl}x^{2}+x+1,&x<1\\x^{2}+2x+3,&1\leq x\leq 2\\x^{3}+2x+3,&x>2\\\end{array}}\right.$
	$\left\lbrace {\begin{array}{rl}x^{2}+x+1,&x<2\\x^{3}+2x+3,&2\leq x\leq 3\\x^{2}+2x+3,&x>3\\\end{array}}\right.$

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 $(-\infty ,0],[0,2],[2,\infty )$. Suppose $g$ is continuous and piecewise linear with different piece definitions on $(-\infty ,1],[1,3],[3,\infty )$. 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 $(-\infty ,0],[0,1],[1,2],[2,3],[3,\infty )$
	It is continuous and piecewise quadratic, with (potentially) different piece definitions on the intervals $(-\infty ,0],[0,1],[1,2],[2,3],[3,\infty )$
	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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