The graph represent the following function translated or reflected:
[tex]f(x)=\log _2x\text{ or }\log _3x[/tex]Lets calculat some values for f(x), considering base 2 or 3:
[tex]\begin{gathered} \text{Base 2:} \\ f(1)=0 \\ f(2)=1 \\ f(4)=2 \\ \text{Base 3:} \\ f(1)=0 \\ f(3)=1 \\ f(9)=2 \end{gathered}[/tex]We can check that both possible functions are defined only at x > 0, and tends to - infinity when x tends to 0
We also can check that the logarithm with base 3 grows slower than the logarithm with base 2
Now, if we take a look at the graph, we will check that there is a vertical asymptote at x = 3 and the graph of function is turned to the left side, and not to the right side, how the graph of a logarithmic function would look like.
Then. the graph represent the function f reflected on the y-axis and translated three units right
We also can check that when x = 2 (which would be equivalent to x = 1 if the function where not reflected or translated), we have y = -1, and not y = 0. Therefore, the function also is translated one unit down.
Whe also can conclude that the graph is related to a logarithmic function with base 2, since when x = 1 (which would be equivalent to x = 2 if the function were not reflected or translated), we have y = 0 (which would be y = 1 if it weren't translated one unit down).
Therefore, the equation for the function represented in the graph must be:
[tex]f(-x+3)-1=\log _2(-x+3)-1_{}[/tex]
