The problem asks you to write a function $h(x)$ in terms of $g(x)$, where $h(x)$ is:

1. Flipped over the x-axis, and
2. Twice as wide as $g(x)$.

Solution:

1. Flipping over the x-axis means multiplying the function by $-1$. So, flipping $g(x)$ would give $-g(x)$.
2. Twice as wide means a horizontal stretch, which is done by replacing $x$ with $\frac{x}{2}$. This stretches the graph horizontally by a factor of 2.

Thus, the function $h(x)$ is given by: $h(x) = -g\left(\frac{x}{2}\right)$

This satisfies both the conditions of flipping over the x-axis and being twice as wide.

Would you like more details or have any further questions?

Relative questions:

1. What happens to the graph of $g(x)$ when it's flipped over the y-axis?
2. How would the function change if it was flipped over both the x-axis and the y-axis?
3. What effect does compressing a function by a factor of 2 have on its graph?
4. How does the transformation $g(kx)$ affect the graph when $k > 1$ or $0 < k < 1$?
5. What happens to the domain and range of $g(x)$ after these transformations?

Tip:

When transforming functions, always check how horizontal and vertical stretches or reflections impact the graph, particularly on the domain and range.