Explanation
We are given the parent function:
[tex]f(x)=\log_x[/tex]
First, the horizontal shift 6 spaces to the left follows the rule:
[tex]f(x)\to f(x+6)[/tex]
Next, the vertical shift 2 spaces down follows the rule:
[tex]\begin{gathered} f(x)\to f(x)-2 \\ f(x+6)\to f(x+6)-2 \end{gathered}[/tex]
Finally, the reflection over y = k can be represented as:
[tex]\begin{gathered} Let\text{ }g(x)\text{ }be\text{ }the\text{ }combined\text{ }function \\ g(x)=2k+2-f(x+6) \\ \therefore g(x)=2k+2-log(x+6) \end{gathered}[/tex]
Suppose k = 2, the graph becomes:
The red curve is the reflected curve.
Hence, the answer is:
[tex]\begin{equation*} g(x)=2k+2-log(x+6) \end{equation*}[/tex]
