Prototype questions - Revision history

MathAdmin at 03:01, 25 February 2015

2015-02-25T03:01:58Z

WikiSysop at 21:57, 23 February 2015

2015-02-23T21:57:54Z

WikiSysop at 21:57, 23 February 2015

2015-02-23T21:57:23Z

MathAdmin: Created page with "2. Find the domain of the following function. Your answer should use interval notation. f(x) = $\displaystyle{\frac{1}{\sqrt{x^2-x-2}}}$ {| class="mw-collapsibl..."

2015-02-23T18:38:35Z

Created page with "2. Find the domain of the following function. Your answer should use interval notation. f(x) = <math> \displaystyle{\frac{1}{\sqrt{x^2-x-2}}} </math> {| class="mw-collapsibl..."

New page

2. Find the domain of the following function. Your answer should use interval notation.
f(x) = <math> \displaystyle{\frac{1}{\sqrt{x^2-x-2}}} </math>

{| class="mw-collapsible mw-collapsed"
! Foundations
|-
|'''The foundations:'''
|-
| What is the domain of g(x) = <math>\frac{1}{x}</math>?
|-
|The function is undefined if the denominator is zero, so x <math>\neq </math> 0.
|-
|Rewriting"x <math>\neq</math> 0" in interval notation( -<math>\infty</math>, 0) <math>\cup</math>(0, <math>\infty</math>)
|-
|What is the domain of h(x) = <math> \sqrt{x} </math>?
|-
|The function is undefined if we have a negative number inside the square root, so x <math>\ge</math> 0
|}

Solution:

{| class = "mw-collapsible mw-collapsed"
! Step 1:
|-
|Factor <math>x^2 - x - 2</math>
|-
|<math> x^2 - x - 2 = (x + 1) (x - 2)</math>
|-
| So we can rewrite f(x) as <math>f(x) = \displaystyle{\frac{1}{\sqrt{(x+1)(x-2)}}}</math>
|}

{|class = "mw-collapsible mw-collapsed"
! Step 2:
|-
|When does the denominator of f(x) = 0?
|-
|<math>\sqrt{(x + 1)(x - 2)} = 0</math>
|-
|(x + 1)(x - 2) = 0
|-
|(x + 1) = 0 or (x - 2) = 0
|-
|x = -1 or x = 2
|-
|So, since the function is undefiend when the denominator is zero, x <math>\neq</math> -1 and x <math>\neq</math> 2
|}

{|class = "mw-collapsible mw-collapsed"
! Step 3:
|-
|What is the domain of <math>h(x) = \sqrt{(x + 1)(x - 2)}</math>
|-
|critical points: x = -1, x = 2
|-
|Test points:
|-
|x = -2: (-2 + 1)(-2 - 2): (-1)(-4) = 4 > 0
|-
|x = 0: (0 + 1)(0 - 2) = -2 < 0
|-
|x = 3: (3 + 1)(3 - 2): 4*1 = 4 > 0
|-
|So the domain of h(x) is <math>(-\infty, -1] \cup [2, \infty)</math>
|}

{|class = "mw-collapsible mw-collapsed"
! Step 4:
|-
|Take the intersection (i.e. common points) of Steps 2 and 3. <math>( - \infty, -1) \cup (2, \infty)</math>
|}

2. Find the domain of the following function. Your answer should use interval notation.
f(x) = <math> \displaystyle{\frac{1}{\sqrt{x^2-x-2}}} </math>

{| class="mw-collapsible mw-collapsed wikitable"
! Foundations
|-
|'''The foundations:'''
|-
| What is the domain of g(x) = <math>\frac{1}{x}</math>?
|-
|The function is undefined if the denominator is zero, so x <math>\neq </math> 0.
|-
|Rewriting"x <math>\neq</math> 0" in interval notation( -<math>\infty</math>, 0) <math>\cup</math>(0, <math>\infty</math>)
|-
|What is the domain of h(x) = <math> \sqrt{x} </math>?
|-
|The function is undefined if we have a negative number inside the square root, so x <math>\ge</math> 0
|}

Solution:

{| class = "mw-collapsible mw-collapsed wikitable"
! Step 1:
|-
|Factor <math>x^2 - x - 2</math>
|-
|<math> x^2 - x - 2 = (x + 1) (x - 2)</math>
|-
| So we can rewrite f(x) as <math>f(x) = \displaystyle{\frac{1}{\sqrt{(x+1)(x-2)}}}</math>
|}

{|class = "mw-collapsible mw-collapsed wikitable"
! Step 2:
|-
|When does the denominator of f(x) = 0?
|-
|<math>\sqrt{(x + 1)(x - 2)} = 0</math>
|-
|(x + 1)(x - 2) = 0
|-
|(x + 1) = 0 or (x - 2) = 0
|-
|x = -1 or x = 2
|-
|So, since the function is undefinend when the denominator is zero, x <math>\neq</math> -1 and x <math>\neq</math> 2
|}

{|class = "mw-collapsible mw-collapsed wikitable"
! Step 3:
|-
|What is the domain of <math>h(x) = \sqrt{(x + 1)(x - 2)}</math>
|-
|critical points: x = -1, x = 2
|-
|Test points:
|-
|x = -2: (-2 + 1)(-2 - 2): (-1)(-4) = 4 > 0
|-
|x = 0: (0 + 1)(0 - 2) = -2 < 0
|-
|x = 3: (3 + 1)(3 - 2): 4*1 = 4 > 0
|-
|So the domain of h(x) is <math>(-\infty, -1] \cup [2, \infty)</math>
|}

{|class = "mw-collapsible mw-collapsed wikitable"
! Step 4:
|-
|Take the intersection (i.e. common points) of Steps 2 and 3. <math>( - \infty, -1) \cup (2, \infty)</math>
|}

2. Find the domain of the following function. Your answer should use interval notation.
<math>f(x) = \displaystyle{\frac{1}{\sqrt{x^2-x-2}}}</math>

{| class="mw-collapsible mw-collapsed"
! Hint 1
|-
| Which x-values lead to division by 0 or square rooting a negative number
|}

{| class="mw-collapsible mw-collapsed"
! Hint 2
|-
| Use a sign chart to determine for which x-values <math>x^2-x-2 > 0</math>
|}

Solution:
{| class="mw-collapsible mw-collapsed"
! Solution
|-
| Since the domain is the collection of x-values for which we don't divide by zero or square root a negative number we want to solve the inequality <math>x^2 - x - 2 > 0</math>
|-
|<math>(x-2)(x+1)>0</math>
|-
|Now we use a sign chart with test numbers -2, 0, and 3
|-
|<math>x = -2: (-2 - 2)(-2 + 1) = (-4)(-1) = 4 > 0</math>
|-
|<math>x = 0: (0 - 2)(0 + 1) = (-2)(1) = -2 < 0</math>
|-
|<math>x = 3: (3 - 2)(3 + 1)= (1)(4) = 4 > 0</math>
|-
|So the solution is <math>(-\infty, -1) \cup (2, \infty)</math>
|}

2. Find the domain of the following function. Your answer should use interval notation.
<math>f(x) = \displaystyle{\frac{1}{\sqrt{x^2-x-2}}}</math>

{| class="mw-collapsible mw-collapsed wikitable"
! Hint 1
|-
| Which x-values lead to division by 0 or square rooting a negative number
|}

{| class="mw-collapsible mw-collapsed wikitable"
! Hint 2
|-
| Use a sign chart to determine for which x-values <math>x^2-x-2 > 0</math>
|}

Solution:
{| class="mw-collapsible mw-collapsed wikitable"
! Solution
|-
| Since the domain is the collection of x-values for which we don't divide by zero or square root a negative number we want to solve the inequality <math>x^2 - x - 2 > 0</math>
|-
|<math>(x-2)(x+1)>0</math>
|-
|Now we use a sign chart with test numbers -2, 0, and 3
|-
|<math>x = -2: (-2 - 2)(-2 + 1) = (-4)(-1) = 4 > 0</math>
|-
|<math>x = 0: (0 - 2)(0 + 1) = (-2)(1) = -2 < 0</math>
|-
|<math>x = 3: (3 - 2)(3 + 1)= (1)(4) = 4 > 0</math>
|-
|So the solution is <math>(-\infty, -1) \cup (2, \infty)</math>
|}

← Older revision		Revision as of 21:57, 23 February 2015
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	\|Take the intersection (i.e. common points) of Steps 2 and 3. <math>( - \infty, -1) \cup (2, \infty)</math>		\|Take the intersection (i.e. common points) of Steps 2 and 3. <math>( - \infty, -1) \cup (2, \infty)</math>
	\|}		\|}
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 f(x) = <math> \displaystyle{\frac{1}{\sqrt{x^2-x-2}}} </math>
-{| class="mw-collapsible mw-collapsed"
+{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 ! Foundations
 |-
@@ Line 21: / Line 21: @@
 Solution:
-{| class = "mw-collapsible mw-collapsed"
+{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
 ! Step 1:
 |-
@@ Line 31: / Line 31: @@
 |}
-{|class = "mw-collapsible mw-collapsed"
+{|class = "mw-collapsible mw-collapsed" style = "text-align:left;"
 ! Step 2:
 |-
@@ Line 47: / Line 47: @@
 |}
-{|class = "mw-collapsible mw-collapsed"
+{|class = "mw-collapsible mw-collapsed" style = "text-align:left;"
 ! Step 3:
 |-
@@ Line 65: / Line 65: @@
 |}
-{|class = "mw-collapsible mw-collapsed"
+{|class = "mw-collapsible mw-collapsed" style = "text-align:left;"
 ! Step 4:
 |-
@@ Line 84: / Line 84: @@
 f(x) = <math> \displaystyle{\frac{1}{\sqrt{x^2-x-2}}} </math>
-{| class="mw-collapsible mw-collapsed wikitable"
+{| class= "wikitable mw-collapsible mw-collapsed"
 ! Foundations
 |-
@@ Line 103: / Line 103: @@
 Solution:
-{| class = "mw-collapsible mw-collapsed wikitable"
+{| class = "mw-collapsible mw-collapsed wikitable" style = "text-align:left;"
 ! Step 1:
 |-
@@ Line 113: / Line 113: @@
 |}
-{|class = "mw-collapsible mw-collapsed wikitable"
+{|class = "mw-collapsible mw-collapsed wikitable" style = "text-align:left;"
 ! Step 2:
 |-
@@ Line 129: / Line 129: @@
 |}
-{|class = "mw-collapsible mw-collapsed wikitable"
+{|class = "mw-collapsible mw-collapsed wikitable" style = "text-align:left;"
 ! Step 3:
 |-
@@ Line 147: / Line 147: @@
 |}
-{|class = "mw-collapsible mw-collapsed wikitable"
+{|class = "mw-collapsible mw-collapsed wikitable" style = "text-align:left;"
 ! Step 4:
 |-
@@ Line 171: / Line 171: @@
 <math>f(x) = \displaystyle{\frac{1}{\sqrt{x^2-x-2}}}</math>
-{| class="mw-collapsible mw-collapsed"
+{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 ! Hint 1
 |-
@@ Line 177: / Line 177: @@
 |}
-{| class="mw-collapsible mw-collapsed"
+{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 ! Hint 2
 |-
@@ Line 184: / Line 184: @@
 Solution:
-{| class="mw-collapsible mw-collapsed"
+{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 ! Solution
 |-
@@ Line 213: / Line 213: @@
 <math>f(x) = \displaystyle{\frac{1}{\sqrt{x^2-x-2}}}</math>
-{| class="mw-collapsible mw-collapsed wikitable"
+{| class="mw-collapsible mw-collapsed wikitable" style = "text-align:left;"
 ! Hint 1
 |-
@@ Line 219: / Line 219: @@
 |}
-{| class="mw-collapsible mw-collapsed wikitable"
+{| class="mw-collapsible mw-collapsed wikitable" style = "text-align:left;"
 ! Hint 2
 |-
@@ Line 226: / Line 226: @@
 Solution:
-{| class="mw-collapsible mw-collapsed wikitable"
+{| class="mw-collapsible mw-collapsed wikitable" style = "text-align:left;"
 ! Solution
 |-

@@ Line 11: / Line 11: @@
 |The function is undefined if the denominator is zero, so x <math>\neq </math> 0.
 |-
-|Rewriting"x <math>\neq</math> 0" in interval notation( -<math>\infty</math>, 0) <math>\cup</math>(0, <math>\infty</math>)
+|Rewriting" <math>x \neq 0</math>" in interval notation( -<math>\infty</math>, 0) <math>\cup</math>(0, <math>\infty</math>)
 |-
 |What is the domain of h(x) = <math> \sqrt{x} </math>?

Prototype questions - Revision history

MathAdmin at 03:01, 25 February 2015

WikiSysop at 21:57, 23 February 2015

WikiSysop at 21:57, 23 February 2015

MathAdmin: Created page with "2. Find the domain of the following function. Your answer should use interval notation. f(x) = \displaystyle{\frac{1}{\sqrt{x^2-x-2}}} {| class="mw-collapsibl..."

MathAdmin: Created page with "2. Find the domain of the following function. Your answer should use interval notation. f(x) = $\displaystyle{\frac{1}{\sqrt{x^2-x-2}}}$ {| class="mw-collapsibl..."