008A Sample Final A, Question 1

Question: Find $f^{-1}(x)$ for $f(x)=\log _{3}(x+3)-1$

Foundations:
1) How would you find the inverse for a simpler function like $f(x)=3x+5$?
2) How do you remove the Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \log _{3}} in the following equation: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \log _{3}} when we write the equation $y=\log _{3}(x)$ we mean y is the number such that $3^{y}=x$

Solution:

Step 1:
We start by replacing f(x) with y.
This leaves us with $y=\log _{3}(x+3)-1$

Step 2:
Now we swap x and y to get $x=\log _{3}(y+3)-1$
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}

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