008A Sample Final A, Question 1

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

Foundations:
1) How would you find the inverse for a simpler function like ${\displaystyle f(x)=3x+5}$?
2) How do you remove the ${\displaystyle \log _{3}}$ in the following equation: ${\displaystyle \log _{3}(x)=y?}$
1) you would replace f(x) by y, switch x and y, and finally solve for y.
2) By the definition of ${\displaystyle \log _{3}}$ when we write the equation ${\displaystyle y=\log _{3}(x)}$ we mean y is the number such that ${\displaystyle 3^{y}=x}$

Solution:

Step 1:
We start by replacing f(x) with y.
This leaves us with ${\displaystyle y=\log _{3}(x+3)-1}$
Step 2:
Now we swap x and y to get ${\displaystyle x=\log _{3}(y+3)-1}$
In the next step we will solve for y.
Step 3:
From ${\displaystyle x=\log _{3}(y+3)-1}$, we add 1 to both sides to get
${\displaystyle x+1=\log _{3}(y+3).}$ Now we will use the relation in Foundations 2) to swap the log for an exponential to get
${\displaystyle y+3=3^{x+1}}$.
Step 4:
After subtracting 3 from both sides we get ${\displaystyle y=3^{x+1}-3}$. Replacing y with ${\displaystyle f^{-1}(x)}$ we arrive at the final answer that
${\displaystyle f^{-1}(x)=3^{x+1}-3}$
${\displaystyle f^{-1}(x)=3^{x+1}-3}$