The domain of a function of f(x) is [-20, -40] and the range is [-7, 15]. Using interval notation, find the domain the range of g(x)=-f(4(x-1))

To find the domain and range of the function \( g(x) = -f(4(x-1)) \), we need to analyze how the transformations applied to \( f(x) \) affect its domain and range.

### 1. **Domain:**
   - The function inside \( f \) is \( 4(x-1) \). 
   - To find the domain of \( g(x) \), set the expression inside \( f \) equal to the domain of \( f(x) \):

   \[
   -20 \leq 4(x-1) \leq -40
   \]

   - Solve the inequality:

     \[
     -20 \leq 4x - 4 \leq -40
     \]

     Add 4 to each part of the inequality:

     \[
     -16 \leq 4x \leq -36
     \]

     Now, divide by 4:

     \[
     -4 \leq x \leq -9
     \]

   So, the domain of \( g(x) \) is \( x \in [-9, -4] \).

### 2. **Range:**
   - The range of \( f(x) \) is given as \([-7, 15]\).
   - The range of \( g(x) = -f(4(x-1)) \) involves reflecting \( f(x) \) over the x-axis due to the negative sign in front of \( f \). 
   - This reflection changes the range from \([-7, 15]\) to \([-15, 7]\), since reflecting over the x-axis flips the values.

### Final Answer:
- **Domain** of \( g(x) \): \([-9, -4]\)
- **Range** of \( g(x) \): \([-15, 7]\)

Would you like more details or have any questions?

Here are 5 related questions:
1. How does shifting a function horizontally affect its domain?
2. What effect does multiplying the input of a function by a constant have on the graph?
3. How does reflecting a function over the x-axis change its range?
4. What is the impact of a vertical shift on the range of a function?
5. Can the domain of a function ever be restricted by a reflection?

**Tip:** Remember that horizontal transformations (shifts and stretches) always affect the domain, while vertical transformations (reflections, stretches, and shifts) primarily affect the range.

Explore how to find the domain and range of the function g(x) = -f(4(x-1)), where f(x) has a domain of [-20, -40] and a range of [-7, 15]. Understand the effects of transformations on the domain and range of functions. Suitable for advanced high school students.

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