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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 $\log _{3}$ when we write the equation Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y=\log _{3}(x)} we mean y is the number such that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 $x=\log _{3}(y+3)-1$
In the next step we will solve for y.

Step 3:
From $x=\log _{3}(y+3)-1$ , we add 1 to both sides to get
$x+1=\log _{3}(y+3).$ Now we will use the relation in Foundations 2) to swap the log for an exponential to get
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y+3=3^{x+1}} .

Step 4:
After subtracting 3 from both sides we get $y=3^{x+1}-3$ . Replacing y with $f^{-1}(x)$ we arrive at the final answer that
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f^{-1}(x)=3^{x+1}-3}

Final Answer:
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f^{-1}(x)=3^{x+1}-3}

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