Difference between revisions of "008A Sample Final A, Question 1"

Revision as of 21:52, 22 May 2015

Question: Find $f^{-1}(x)$ 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:
Starting 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 x = \log_3(y + 3) - 1} , we start by adding 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}} . All we have to do is subtract 3 from both sides to yield the final answer

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}

Return to Sample Exam

@@ Line 7: / Line 7: @@
 |1) How would you find the inverse for a simpler function like <math>f(x) = 3x + 5</math>?
 |-
-|2) How are <math>log_3(x)</math> and <math>3^x</math> related?
+|2) How do you remove the <math>\log_3</math> in the following equation: <math>\log_3(x) = y?</math>
 |-
 |Answers:
@@ Line 13: / Line 13: @@
 |1) you would replace f(x) by y, switch x and y, and finally solve for y.
 |-
-|2) By stating <math>y = \log_3(x)</math> we also get the following relation <math>x = 3^y</math>
+|2) By the definition of <math>\log_3</math> when we write the equation <math>y = \log_3(x)</math> we mean y is the number such that <math>3^y = x</math>
 |}

Difference between revisions of "008A Sample Final A, Question 1"

Revision as of 21:52, 22 May 2015

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