Let's break down and solve the task based on the image:

Task 1: Identify the correct graph of the function.

You are asked to choose the correct graph of a function. Unfortunately, without seeing the function's equation (which isn't provided), it's hard to choose the right one. However, we can describe the curves visible:

• There are four graphs labeled A, B, C, D.
• Each graph represents a different transformation of what seems to be a logarithmic function.

We could make a better guess based on further clues, or if you have any specific functions involved.

Task 2: Find the equation of the inverse function (g(x)) from the graph.

The second part asks to determine the inverse function based on the given options.

Here are some observations about the inverse options:

• Logarithmic and root functions are inverse to exponentials and quadratic functions, respectively.
• The general form of a logarithmic inverse is $g(x) = \log_b(x)$ for some base $b$, or some transformation of a square root like $g(x) = \sqrt{x}$.

Some of the provided options are:

• $g(x) = \frac{1}{3}\sqrt{x + 3} - 4$
• $g(x) = \log_3(x - 3) - 4$
• $g(x) = 3^{(x + 4)} - 3$

These choices suggest you're dealing with an inverse that involves shifting and rescaling the axes. You should focus on whether the original graph is exponential, logarithmic, or a square root function.

To proceed, could you clarify the function you're working with or give me more details about what equation you're looking at? This way, I can help you identify the graph and inverse correctly.

Would you like any detailed explanation or assistance with these specific transformations or the behavior of inverses?

Relative Questions:

1. What is the original equation of the function whose inverse you're finding?
2. Are the graphs representing logarithmic, exponential, or root functions?
3. What transformations (shifts, scaling) are applied to the function or its inverse?
4. How does identifying the graph help find the inverse?
5. What are common forms of inverse functions for exponential and logarithmic equations?

Tip:

When finding the inverse of a function graphically, switch the $x$ and $y$-coordinates and analyze the symmetry about the line $y = x$. This can help visually identify the correct graph.