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Let's go over each part of the image and solve the problems.

### Problem 5
The graph of \( f(x) \) is provided, and we need to find the **domain** and **range**.

- **Domain**: The domain is the set of all possible input values (x-values) for which the function is defined. From the graph, the function is defined for \( x \) values between \(-5\) and \(5\), inclusive.

  \[
  \text{Domain: } [-5, 5]
  \]

- **Range**: The range is the set of all possible output values (y-values) that the function can take. From the graph, the lowest point on the graph is at \( y = -3 \) and the highest point is at \( y = 3 \).

  \[
  \text{Range: } [-3, 3]
  \]

---

### Problem 6
We are given the function \( h(x) = -2f(x + 3) - 1 \) and need to find the **domain** and **range**.

- **Domain**: The transformation \( x + 3 \) is a horizontal shift 3 units to the left. Therefore, the domain of \( h(x) \) is the same as the domain of \( f(x) \), shifted by 3 units to the left. The domain of \( f(x) \) was \( [-5, 5] \), so shifting this to the left by 3 gives:

  \[
  \text{Domain: } [-8, 2]
  \]

- **Range**: The transformation involves multiplying the function by \(-2\), which reflects it over the x-axis and stretches it by a factor of 2. Additionally, subtracting 1 shifts the graph down by 1 unit. The range of \( f(x) \) was \( [-3, 3] \). After reflecting and stretching, this becomes \( [-6, 6] \). Finally, subtracting 1 gives:

  \[
  \text{Range: } [-7, -5]
  \]

---

### Problem 7
The function \( k(x) \) is constructed by applying three transformations to \( h(x) \):

- Horizontal dilation by a factor of 4,
- Vertical dilation by a factor of \( \frac{1}{2} \),
- Vertical translation by 3 units.

The general form of \( k(x) \) is given as \( k(x) = ah(bx) + c \), and we need to find the values of \( a \), \( b \), and \( c \).

- Horizontal dilation by 4 means \( b = \frac{1}{4} \).
- Vertical dilation by \( \frac{1}{2} \) means \( a = \frac{1}{2} \).
- Vertical translation by 3 means \( c = 3 \).

Thus, the values are:
\[
a = \frac{1}{2}, \quad b = \frac{1}{4}, \quad c = 3
\]

---

Would you like any further explanation or details?

Here are five related questions to deepen understanding:
1. How would the range change if \( f(x) \) had a higher maximum point on the graph?
2. What happens to the domain if \( h(x) \) had a transformation like \( f(2x) \)?
3. How does reflecting a function over the y-axis affect its domain and range?
4. Can you describe how to combine multiple transformations in a single step?
5. What is the effect of multiplying a function by a negative constant?

**Tip**: When working with transformations, it's helpful to apply one transformation at a time and see its effect on both the domain and range.

The image includes problems about finding domains and ranges of functions and applying transformations to graphs.

Explore how to find domains and ranges of functions, apply function transformations like dilations, shifts, and reflections with detailed solutions. Ideal for high school students, Grades 9-12.

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