Quiz:Piecewise definition of function: Difference between revisions

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 ==Pointwise combination==
-{{fillin}}
+<quiz display=simple>
+{Suppose <math>f(x) := \left\lbrace \begin{array}{rl} x^2, & x < 2, x^3, & x \ge 2 \\\end{array}\right.</math> and <math>g(x) := \left \lbrace \begin{array}{rl} x + 1, & x \le 1, 2x + 3, & x > 1 \\\end{array}\right.</math>. What is <math>(f + g)(x)</math>?
+|type="()"}
+- <math>\left \lbrace\begin{array}{rl} x^2 + x + 1, & x \le 3 \\ 3x + 3, & x > 3 \\\end{array}\right.</math>
+- <math>\left \lbrace\begin{array}{rl} x^2 + x + 1, & x < 3 \\ 3x + 3, & x \ge 3 \\\end{array}\right.</math>
++ <math>\left \lbrace\begin{array}{rl} x^2 + x + 1, & x \le 1 \\ x^2 + 2x + 3, & 1 < x < 2 \\ x^3 + 2x + 3, & x \ge 2 \\\end{array}\right.</math>
+- <math>\left \lbrace\begin{array}{rl} x^2 + x + 1, & x < 1 \\ x^2 + 2x + 3, & 1 \le x \le 2 \\ x^3 + 2x + 3, & x > 2 \\\end{array}\right.</math>
+- <math>\left \lbrace\begin{array}{rl} x^2 + x + 1, & x < 2 \\ x^3 + 2x + 3, & 2 \le x \le 3 \\ x^2 + 2x + 3, & x > 3 \\\end{array}\right.</math>
-==Continuity and one-sided continuity==
+==Continuity and pointwise combination==
-{{fillin}}
+<quiz display=simple>
+{Suppose <math>f</math> and <math>g</math> are functions defined on a closed interval <math>[a,b]</math> and are both piecewise continuous, i.e., each function is continuous except possibly at finitely many points in the open interval <math>(a,b)</math>. Then, which of the following functions is ''not'' guaranteed to be a piecewise continuous function on <math>[a,b]</math>?
+|type="()"}
+- <math>\! f + g</math>, the [[pointwise sum of functions]]
+- <math>\! f - g</math>, the [[pointwise difference of functions]]
+- <math>\! f \cdot g</math>, the [[pointwise product of functions]]
++ None of the above, i.e., they are ''all'' guaranteed to be piecewise continuous functions on <math>[a,b]</math>
+- All of the above, i.e., none of them is guaranteed to be a piecewise continuous function on <math>[a,b]</math>
+{Suppose <math>f</math> and <math>g</math> are functions defined on all of <math>\R</math>. <math>f</math> is discontinuous at 5 points and <math>g</math> is discontinuous at 3 points. What can we say about <math>f + g</math>?
+|type="()"}
+- 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.
+{Suppose <math>f</math> and <math>g</math> are functions defined on all of <math>\R</math>. <math>f</math> is discontinuous at 5 points and <math>g</math> is discontinuous at 3 points. What can we say about <math>f \cdot g</math>?
+|type="()"}
+- 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
+{Suppose <math>f</math> and <math>g</math> are functions defined on all of <math>\R</math>. Suppose <math>f</math> is continuous and piecewise linear, with different nonconstant linear piece definitions on the interval <math>(-\infty,0],[0,2],[2,\infty)</math>. Suppose <math>g</math> is continuous and piecewise linear with different piece definitions on <math>(-\infty,1],[1,3],[3,\infty)</math>. What can we say about the [[pointwise product of functions]] <math>f \cdot g</math>?
+|type="()"}
+- It is continuous and piecewise linear, with (potentially) different piece definitions on the intervals <math>(-\infty,0],[0,1],[1,2],[2,3],[3,\infty)</math>
++ It is continuous and piecewise quadratic, with (potentially) different piece definitions on the intervals <math>(-\infty,0],[0,1],[1,2],[2,3],[3,\infty)</math>
+- 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
+</quiz>
 ==Composition==

	${\begin{matrix} x^{2} + x + 1, & x \leq 3 \\ 3 x + 3, & x > 3 \end{matrix}$
	${\begin{matrix} x^{2} + x + 1, & x < 3 \\ 3 x + 3, & x \geq 3 \end{matrix}$
	${\begin{matrix} x^{2} + x + 1, & x \leq 1 \\ x^{2} + 2 x + 3, & 1 < x < 2 \\ x^{3} + 2 x + 3, & x \geq 2 \end{matrix}$
	${\begin{matrix} x^{2} + x + 1, & x < 1 \\ x^{2} + 2 x + 3, & 1 \leq x \leq 2 \\ x^{3} + 2 x + 3, & x > 2 \end{matrix}$
	${\begin{matrix} x^{2} + x + 1, & x < 2 \\ x^{3} + 2 x + 3, & 2 \leq x \leq 3 \\ x^{2} + 2 x + 3, & x > 3 \end{matrix}$
	$f + g$ , 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 $[a, b]$

	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.

	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

	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

Quiz:Piecewise definition of function: Difference between revisions

Revision as of 21:47, 19 October 2011

Pointwise combination

Composition

Differentiation