Difference between revisions of "005 Sample Final A, Question 3"

Revision as of 19:12, 21 May 2015

Question Find f 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 \circ} g and its domain if 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) = x^2+1 \qquad g(x)=\sqrt{x-1}}

Foundations:
1) How do you compose two functions, such as given 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^2} and 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 g = x + 1} , what is fFailed 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 \circ} g?
2) When should a point x be in the domain of fFailed 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 \circ} g?
Answers:
1) We replace all occurrences of x in f with g, so $f\circ g=(x+1)^{2}$ .
2) A point should be in the domain of fFailed 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 \circ} g when it is in the domain of g, and g(x) is in the domain of f.

Step 1:

First we find the domain of g. Since f 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 \circ} g = f(g(x)). So if x is not in the domain of g, it is not in the domain of f 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 \circ} g. The domain of g 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 [1, \infty)} .

Step 2:

To find f 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 \circ} g we replace any occurrence of x in f with g, to yield 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})^2 + 1 = x - 1 + 1 = x }

Final Answers:

f 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 \circ} g = 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 } , and the domain 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 [1, \infty)} .

@@ Line 3: / Line 3: @@
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-! Step 1
+! Foundations:
+|-
+|1) How do you compose two functions, such as given <math>f = x^2</math>&nbsp; and &nbsp; <math>g = x + 1</math>, what is f<math>\circ</math>g?
+|-
+|2) When should a point x be in the domain of f<math>\circ</math>g?
+|-
+|Answers:
+|-
+|1) We replace all occurrences of x in f with g, so <math>f\circ g = (x + 1)^2</math>.
+|-
+|2) A point should be in the domain of f<math>\circ</math>g when it is in the domain of g, and g(x) is in the domain of f.
+|}
+{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+! Step 1:
 |-
 |First we find the domain of g. Since f <math>\circ</math> g = f(g(x)). So if x is not in the domain of g, it is not in the domain of f <math>\circ</math> g. The domain of g is <math>[1, \infty)</math>.
@@ Line 9: / Line 23: @@
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-! Step 2
+! Step 2:
 |-
 |To find f <math>\circ</math> g we replace any occurrence of x in f with g, to yield <math>(\sqrt{x - 1})^2 + 1 = x - 1 + 1 = x </math>
@@ Line 15: / Line 29: @@
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-! Final Answers
+! Final Answers:
 |-
 |f <math>\circ</math> g = <math> x </math>, and the domain is <math>[1, \infty)</math>.
 |}

Difference between revisions of "005 Sample Final A, Question 3"

Revision as of 19:12, 21 May 2015

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