The image shows a math problem with functions $f(x) = 4x - 1$ and $g(x) = -7x^2$. You are asked to:

1. Find the domain of $f$, $g$, $f+g$, $f-g$, $fg$, $\frac{f}{g}$, and $\frac{g}{f}$.
2. Find and simplify expressions for:
   • $(fg)(x)$
   • $(ff)(x)$
   • $\frac{f}{g}(x)$

Let's start solving:

a) Domain of $f(x)$ and $g(x)$:

• $f(x) = 4x - 1$: This is a linear function, so its domain is $\mathbb{R}$ (all real numbers).
• $g(x) = -7x^2$: This is a quadratic function, so its domain is also $\mathbb{R}$.

For the domain of combined functions:

• $f+g$, $f-g$, and $fg$ all have domain $\mathbb{R}$, since both functions are defined for all real numbers.
• $\frac{f}{g}$ and $\frac{g}{f}$ require that the denominator is not zero, so:
  • For $\frac{f}{g}$, $g(x) \neq 0$, which gives $-7x^2 \neq 0$, so $x \neq 0$. The domain is $\mathbb{R} \setminus \{0\}$.
  • For $\frac{g}{f}$, $f(x) \neq 0$, which gives $4x - 1 \neq 0$, so $x \neq \frac{1}{4}$. The domain is $\mathbb{R} \setminus \left\{ \frac{1}{4} \right\}$.

b) Finding the expressions:

1. $(fg)(x) = f(x) \cdot g(x)$: $(fg)(x) = (4x - 1)(-7x^2) = -28x^3 + 7x^2$

2. $(ff)(x) = f(x) \cdot f(x)$: $(ff)(x) = (4x - 1)^2 = 16x^2 - 8x + 1$

3. $\frac{f}{g}(x) = \frac{f(x)}{g(x)}$: $\frac{f}{g}(x) = \frac{4x - 1}{-7x^2}$

Let me know if you'd like a further breakdown or more details on any of these steps.

Here are 5 related questions to expand your understanding:

1. What is the domain of a rational function in general?
2. How do you find the range of a quadratic function like $g(x) = -7x^2$?
3. Can you factor $(ff)(x)$ further? How does factoring help in simplifying expressions?
4. What happens to the domain if we change $f(x) = 4x - 1$ to a different form, such as a rational or exponential function?
5. How can you verify the simplified expression $(fg)(x) = -28x^3 + 7x^2$ by substitution or graphing?

Tip: Always check the domain of functions before simplifying, especially for rational functions where denominators can introduce restrictions.