Difference between revisions of "Prototype questions"
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f(x) = <math> \displaystyle{\frac{1}{\sqrt{x^2-x-2}}} </math> | 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 | ! Foundations | ||
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Solution: | Solution: | ||
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! Step 4: | ! Step 4: | ||
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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> | f(x) = <math> \displaystyle{\frac{1}{\sqrt{x^2-x-2}}} </math> | ||
| − | {| class="mw-collapsible mw-collapsed | + | {| class= "wikitable mw-collapsible mw-collapsed" |
! Foundations | ! Foundations | ||
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Solution: | Solution: | ||
| − | {| class = "mw-collapsible mw-collapsed wikitable" | + | {| class = "mw-collapsible mw-collapsed wikitable" style = "text-align:left;" |
! Step 1: | ! Step 1: | ||
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! Step 2: | ! Step 2: | ||
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| − | {|class = "mw-collapsible mw-collapsed wikitable" | + | {|class = "mw-collapsible mw-collapsed wikitable" style = "text-align:left;" |
! Step 3: | ! Step 3: | ||
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| − | {|class = "mw-collapsible mw-collapsed wikitable" | + | {|class = "mw-collapsible mw-collapsed wikitable" style = "text-align:left;" |
! Step 4: | ! Step 4: | ||
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<math>f(x) = \displaystyle{\frac{1}{\sqrt{x^2-x-2}}}</math> | <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 | ! Hint 1 | ||
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| − | {| class="mw-collapsible mw-collapsed" | + | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" |
! Hint 2 | ! Hint 2 | ||
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Solution: | Solution: | ||
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! Solution | ! Solution | ||
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<math>f(x) = \displaystyle{\frac{1}{\sqrt{x^2-x-2}}}</math> | <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 | ! Hint 1 | ||
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| − | {| class="mw-collapsible mw-collapsed wikitable" | + | {| class="mw-collapsible mw-collapsed wikitable" style = "text-align:left;" |
! Hint 2 | ! Hint 2 | ||
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Solution: | Solution: | ||
| − | {| class="mw-collapsible mw-collapsed wikitable" | + | {| class="mw-collapsible mw-collapsed wikitable" style = "text-align:left;" |
! Solution | ! Solution | ||
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Latest revision as of 19:01, 24 February 2015
2. Find the domain of the following function. Your answer should use interval notation. f(x) = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle{\frac{1}{\sqrt{x^2-x-2}}} }
| Foundations |
|---|
| The foundations: |
| What is the domain of g(x) = ? |
| The function is undefined if the denominator is zero, so x Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \neq } 0. |
| Rewriting" Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \neq 0} " in interval notation( -Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \infty} , 0) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cup} (0, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \infty} ) |
| What is the domain of h(x) = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{x} } ? |
| The function is undefined if we have a negative number inside the square root, so x Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ge} 0 |
Solution:
| Step 1: |
|---|
| Factor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2 - x - 2} |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2 - x - 2 = (x + 1) (x - 2)} |
| So we can rewrite f(x) as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \displaystyle{\frac{1}{\sqrt{(x+1)(x-2)}}}} |
| Step 2: |
|---|
| When does the denominator of f(x) = 0? |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{(x + 1)(x - 2)} = 0} |
| (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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \neq} -1 and x Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \neq} 2 |
| Step 3: |
|---|
| What is the domain of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(x) = \sqrt{(x + 1)(x - 2)}} |
| 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-\infty, -1] \cup [2, \infty)} |
| Step 4: |
|---|
| Take the intersection (i.e. common points) of Steps 2 and 3. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ( - \infty, -1) \cup (2, \infty)} |
2. Find the domain of the following function. Your answer should use interval notation. f(x) = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle{\frac{1}{\sqrt{x^2-x-2}}} }
| Foundations |
|---|
| The foundations: |
| What is the domain of g(x) = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{x}} ? |
| The function is undefined if the denominator is zero, so x Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \neq } 0. |
| Rewriting"x Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \neq} 0" in interval notation( -Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \infty} , 0) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cup} (0, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \infty} ) |
| What is the domain of h(x) = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{x} } ? |
| The function is undefined if we have a negative number inside the square root, so x Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ge} 0 |
Solution:
| Step 1: |
|---|
| Factor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2 - x - 2} |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2 - x - 2 = (x + 1) (x - 2)} |
| So we can rewrite f(x) as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \displaystyle{\frac{1}{\sqrt{(x+1)(x-2)}}}} |
| Step 2: |
|---|
| When does the denominator of f(x) = 0? |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{(x + 1)(x - 2)} = 0} |
| (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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \neq} -1 and x Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \neq} 2 |
| Step 3: |
|---|
| What is the domain of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(x) = \sqrt{(x + 1)(x - 2)}} |
| 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-\infty, -1] \cup [2, \infty)} |
| Step 4: |
|---|
| Take the intersection (i.e. common points) of Steps 2 and 3. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ( - \infty, -1) \cup (2, \infty)} |
2. Find the domain of the following function. Your answer should use interval notation.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \displaystyle{\frac{1}{\sqrt{x^2-x-2}}}}
| Hint 1 |
|---|
| Which x-values lead to division by 0 or square rooting a negative number |
| Hint 2 |
|---|
| Use a sign chart to determine for which x-values Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2-x-2 > 0} |
Solution:
| 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2 - x - 2 > 0} |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x-2)(x+1)>0} |
| Now we use a sign chart with test numbers -2, 0, and 3 |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = 0: (0 - 2)(0 + 1) = (-2)(1) = -2 < 0} |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = 3: (3 - 2)(3 + 1)= (1)(4) = 4 > 0} |
| So the solution is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-\infty, -1) \cup (2, \infty)} |
2. Find the domain of the following function. Your answer should use interval notation. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \displaystyle{\frac{1}{\sqrt{x^2-x-2}}}}
| Hint 1 |
|---|
| Which x-values lead to division by 0 or square rooting a negative number |
| Hint 2 |
|---|
| Use a sign chart to determine for which x-values Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2-x-2 > 0} |
Solution:
| 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2 - x - 2 > 0} |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x-2)(x+1)>0} |
| Now we use a sign chart with test numbers -2, 0, and 3 |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = -2: (-2 - 2)(-2 + 1) = (-4)(-1) = 4 > 0} |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = 0: (0 - 2)(0 + 1) = (-2)(1) = -2 < 0} |
| So the solution is |
