The question is asking to determine whether the functions $f(x)$ and $g(x)$ are invertible. To determine this, we need to check if both functions pass the Horizontal Line Test. A function is invertible if and only if every horizontal line intersects the graph of the function at most once (which guarantees the function is one-to-one).

1. Function $f(x)$

• Domain: $(-\infty, \infty)$
• Range: $[2.75, \infty)$

From the graph, $f(x)$ appears to be a decreasing function. Since $f(x)$ does not repeat any $y$-value (within its given range), it passes the horizontal line test. Therefore, $f(x)$ is invertible.

2. Function $g(x)$

• Domain: $(0, \infty)$
• Range: $(-\infty, \infty)$

The graph of $g(x)$ is a continuous increasing function. It never repeats any $y$-value as it increases continuously. Hence, $g(x)$ also passes the horizontal line test. Therefore, $g(x)$ is invertible as well.

Conclusion:

Both $f(x)$ and $g(x)$ are invertible functions based on the horizontal line test.

Would you like further details on how to compute the inverse of these functions or any other explanations?

Here are some relative questions that expand on this concept:

1. How do we algebraically find the inverse of a function?
2. What are the graphical properties of an inverse function?
3. Can all one-to-one functions be inverted?
4. How does the domain and range of a function change when finding its inverse?
5. What happens if a function fails the horizontal line test?

Tip: When verifying if a function is invertible, always consider the behavior of the function over its entire domain and range.