The problem asks to write a function $g(x)$ whose graph represents a reflection in the $y$-axis of the graph of $f(x) = \ln(x) - 2$.

Solution:

To reflect a function across the $y$-axis, you replace $x$ with $-x$ in the function. So, for the function $f(x) = \ln(x) - 2$, reflecting it across the $y$-axis gives:

$g(x) = f(-x) = \ln(-x) - 2$

Thus, the reflected function is:

$g(x) = \ln(-x) - 2$

Note that this new function, $g(x) = \ln(-x) - 2$, is only defined for $x < 0$, since the natural logarithm, $\ln(x)$, is only defined for positive values of $x$, and $-x$ must be positive.

Would you like any further details or have questions?

Relative Questions:

1. What is the domain of the function $g(x) = \ln(-x) - 2$?
2. How does the reflection of a function affect its graph visually?
3. What is the inverse of $f(x) = \ln(x) - 2$?
4. How would you translate $f(x) = \ln(x) - 2$ instead of reflecting it?
5. What happens if you reflect the function across the $x$-axis instead?

Tip:

When reflecting a function across the $y$-axis, remember to replace $x$ with $-x$ in the equation of the function.