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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.

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**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.

The functions f and g are defined as f(x) = 4x - 1 and g(x) = -7x^2. Find the domain of f, g, f + g, f - g, fg, f/g, and g/f. Then find (fg)(x), (ff)(x), and f/g(x), simplifying your answers.

Explore how to find the domain and perform operations on functions f(x) = 4x - 1 and g(x) = -7x^2. This problem covers finding the domain of various combinations of functions, along with simplifying expressions for (fg)(x), (ff)(x), and f/g(x). Suitable for students in Grades 10-12.

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