Difference between revisions of "008A Sample Final A, Question 1"
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;" | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| − | ! Foundations: | + | ! Foundations: |
|- | |- | ||
|1) How would you find the inverse for a simpler function like <math>f(x) = 3x + 5</math>? | |1) How would you find the inverse for a simpler function like <math>f(x) = 3x + 5</math>? | ||
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{| class = "mw-collapsible mw-collapsed" style = "text-align:left;" | {| class = "mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| − | ! Step 1: | + | ! Step 1: |
|- | |- | ||
|We start by replacing f(x) with y. | |We start by replacing f(x) with y. | ||
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{|class = "mw-collapsible mw-collapsed" style = "text-align:left;" | {|class = "mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| − | ! Step 2: | + | ! Step 2: |
|- | |- | ||
|Now we swap x and y to get <math>x = \log_3(y + 3) - 1</math> | |Now we swap x and y to get <math>x = \log_3(y + 3) - 1</math> | ||
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{|class = "mw-collapsible mw-collapsed" style = "text-align:left;" | {|class = "mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| − | ! Step 3: | + | ! Step 3: |
|- | |- | ||
|From <math>x = \log_3(y + 3) - 1</math>, we add 1 to both sides to get | |From <math>x = \log_3(y + 3) - 1</math>, we add 1 to both sides to get | ||
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{|class = "mw-collapsible mw-collapsed" style = "text-align:left;" | {|class = "mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| − | ! Step 4: | + | ! Step 4: |
|- | |- | ||
|After subtracting 3 from both sides we get <math>y = 3^{x+1}-3</math>. Replacing y with <math>f^{-1}(x)</math> we arrive at the final answer that | |After subtracting 3 from both sides we get <math>y = 3^{x+1}-3</math>. Replacing y with <math>f^{-1}(x)</math> we arrive at the final answer that | ||
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{|class = "mw-collapsible mw-collapsed" style = "text-align:left;" | {|class = "mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| − | ! Final Answer: | + | ! Final Answer: |
|- | |- | ||
|<math>f^{-1}(x) = 3^{x+1} - 3</math> | |<math>f^{-1}(x) = 3^{x+1} - 3</math> | ||
Revision as of 22:49, 25 May 2015
Question: Find for 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) = \log_3(x+3)-1}
| Foundations: |
|---|
| 1) How would you find the inverse for a simpler function like 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) = 3x + 5} ? |
| 2) How do you remove the 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 \log_3} in the following equation: 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 \log_3(x) = y?} |
| Answers: |
| 1) you would replace f(x) by y, switch x and y, and finally solve for y. |
| 2) By the definition 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 \log_3} when we write the equation 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 y = \log_3(x)} we mean y is the number such that 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 3^y = x} |
Solution:
| Step 1: |
|---|
| We start by replacing f(x) with y. |
| This leaves us with 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 y = \log_3(x + 3) - 1} |
| Step 2: |
|---|
| Now we swap x and y to get 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 = \log_3(y + 3) - 1} |
| In the next step we will solve for y. |
| Step 3: |
|---|
| From 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 = \log_3(y + 3) - 1} , we add 1 to both sides to get |
| 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 + 1 = \log_3(y + 3).} Now we will use the relation in Foundations 2) to swap the log for an exponential to get |
| 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 y + 3 = 3^{x+1}} . |
| Step 4: |
|---|
| After subtracting 3 from both sides we get 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 y = 3^{x+1}-3} . Replacing y with 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^{-1}(x)} we arrive at the final answer that |
| 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^{-1}(x) = 3^{x+1} - 3} |
| Final Answer: |
|---|
| 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^{-1}(x) = 3^{x+1} - 3} |