The image contains a math assignment requiring students to compute combinations of functions $f(x)$ and $g(x)$, specifically:

1. $(f + g)(x) = f(x) + g(x)$
2. $(f - g)(x) = f(x) - g(x)$
3. $(f \cdot g)(x) = f(x) \cdot g(x)$
4. $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$, with the domain excluding values where $g(x) = 0$.

The tasks are for the following pairs of functions:

2.

• $f(x) = x - 1$
• $g(x) = 5x - 2$

4.

• $f(x) = x^2$
• $g(x) = x - 5$

6.

• $f(x) = 3x^2 - 4$
• $g(x) = x^2 - 8x + 4$

I will compute the required operations for problem 2 first and show detailed steps. Let me know if you'd like solutions for all the exercises! sympy import symbols, simplify

Define the variable and functions for problem 2

x = symbols('x') f_2 = x - 1 g_2 = 5 * x - 2

Perform operations

f_plus_g_2 = simplify(f_2 + g_2) f_minus_g_2 = simplify(f_2 - g_2) f_times_g_2 = simplify(f_2 * g_2) f_divided_by_g_2 = simplify(f_2 / g_2)

f_plus_g_2, f_minus_g_2, f_times_g_2, f_divided_by_g_2For problem 2, where $f(x) = x - 1$ and $g(x) = 5x - 2$, the computations are:

1. $(f + g)(x) = f(x) + g(x) = 6x - 3$
2. $(f - g)(x) = f(x) - g(x) = 1 - 4x$
3. $(f \cdot g)(x) = f(x) \cdot g(x) = (x - 1)(5x - 2)$
4. $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{x - 1}{5x - 2}$, where $g(x) \neq 0$.

Would you like me to expand or solve for specific values? Should I proceed with problem 4 or 6?

Here are some follow-up questions to expand on this:

1. What happens to the domain of $(f/g)(x)$ when $g(x)$ equals zero?
2. Can you simplify $(f \cdot g)(x)$ fully into standard polynomial form?
3. What are the x-intercepts of $f(x)$, $g(x)$, and $f+g(x)$?
4. What happens when you substitute $x = 1$ in all the computed expressions?
5. How does the behavior of $f(x)$ and $g(x)$ compare as $x \to \infty$?

Tip:

When dividing two functions, always check for values of $x$ that make the denominator zero, as they will be excluded from the domain.