8A F11 Q1

Question: Find 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)} 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 are 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)} 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 3^x} related?
Answers:
1) you would replace f(x) by y, switch x and y, and finally solve for y.
2) By stating 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 also get the following relation 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 = 3^y}

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:
Starting with $x=\log _{3}(y+3)-1$, we start by adding 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
$y+3=3^{x+1}$. 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 $y=3^{x+1}-3$. Replacing y with $f^{-1}(x)$ we arrive at the final answer that
$f^{-1}(x)=3^{x+1}-3$

Final Answer:
$f^{-1}(x)=3^{x+1}-3$

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