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 are $log_{3}(x)$ and $3^{x}$ related?
Answers:
1) you would replace f(x) by y, switch x and y, and finally solve for y.
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 $x=\log _{3}(y+3)-1$, 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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