Let's work through each of these math problems step by step, focusing first on the primary question.

Question 9: Write $g(x)$ as a transformation of $f(x)$

We are given two functions $f(x)$ and $g(x)$, whose graphs are shown in the first image (file-1). Let's observe how $g(x)$ is derived from $f(x)$.

Analysis:

• $f(x)$ is the red graph, and it appears to be an increasing function, with a horizontal asymptote at $y = 5$ and $y = -5$.
• $g(x)$, the blue graph, is essentially the reflection of $f(x)$, but flipped across both the y-axis and the x-axis.

This suggests that $g(x)$ is a reflection of $f(x)$ across the origin.

Transformation:

To reflect a function across both axes, we apply both horizontal and vertical reflections. This can be written as:

$g(x) = -f(-x)$

Thus, $g(x)$ is $f(x)$, but with both the x and y axes reflected.

Would you like to go over any details of this transformation or clarify other parts of the image? Here are 5 related questions to consider:

1. What would happen if we only reflected the function across the x-axis?
2. Can you describe how $f(x)$ would change if we translated it instead of reflecting it?
3. How does the reflection across the y-axis differ from the reflection across the x-axis?
4. What is the general form of a transformation involving scaling and reflecting?
5. How do vertical and horizontal shifts affect the graph of a function?

Tip: When identifying transformations from graphs, check for shifts, reflections, stretches, or compressions based on how the two functions relate geometrically.