# 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 do you remove the $\log _{3}$ in the following equation: $\log _{3}(x)=y?$ 1) you would replace f(x) by y, switch x and y, and finally solve for y.
2) By the definition of $\log _{3}$ when we write the equation $y=\log _{3}(x)$ we mean y is the number such that $3^{y}=x$ 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:
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
$y+3=3^{x+1}$ .
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
$f^{-1}(x)=3^{x+1}-3$ $f^{-1}(x)=3^{x+1}-3$ 