8A F11 Q1

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 are $log_{3}(x)$ and $3^{x}$ related?
For 1) you would replace f(x) by y, switch x and y, and finally solve for y.
For 2) By stating $y=\log _{3}(x)$ we also get the following relation $x=3^{y}$

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