The image contains a set of problems asking to graph functions using the given $f(x)$, and then apply transformations to find $g(x) = |f(x)|$ and $h(x) = f(|x|)$.

Here’s how you approach each:

Problem 41: $f(x) = \frac{2}{x}$

• Graph $f(x)$: This is a hyperbola, with vertical and horizontal asymptotes at $x = 0$ and $y = 0$.
• Graph $g(x) = |f(x)|$: This reflects the negative part of $f(x)$ (where $f(x) < 0$) above the x-axis.
• Graph $h(x) = f(|x|)$: Reflect the graph of $f(x)$ to the right of the y-axis onto the left side (making it symmetric).

Problem 42: $f(x) = \sqrt{x^2 - 4}$

• Graph $f(x)$: This is a square root function with horizontal shifts. It starts at $x = -2$ and $x = 2$ and goes outward.
• Graph $g(x) = |\sqrt{x^2 - 4}|$: Since $f(x)$ is non-negative, this is the same as the original function.
• Graph $h(x) = f(|x|)$: Reflect the graph over the y-axis to make it symmetric.

Problem 43: $f(x) = x^4 - x^3 - 4x^2$

• Graph $f(x)$: This is a quartic function. Identify key features like critical points and inflection points.
• Graph $g(x) = |f(x)|$: Reflect the parts of the graph where $f(x) < 0$ above the x-axis.
• Graph $h(x) = f(|x|)$: Reflect the right side of the graph onto the left side to make it symmetric.

Problem 44: $f(x) = \frac{1}{2}x^3 + 2x^2 - 8x - 2$

• Graph $f(x)$: This is a cubic polynomial. Find the inflection points and turning points for sketching.
• Graph $g(x) = |f(x)|$: Reflect the negative parts of the graph above the x-axis.
• Graph $h(x) = f(|x|)$: Reflect the right side of the graph onto the left side for symmetry.

Would you like detailed step-by-step explanations for any of these graphs?

Here are five related questions:

1. How does reflecting across the x-axis differ from reflecting across the y-axis in graphing?
2. How can you determine if a function is even or odd by looking at its graph?
3. What is the effect of the absolute value on the graph of a function?
4. What are some common transformations applied to graphs of functions (shifts, stretches, reflections)?
5. How does graphing cubic and quartic functions differ in terms of identifying key points?

Tip: When graphing absolute value transformations, always start by sketching the original function to understand where reflections occur!