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Let's break down and solve each of the five functions given in the worksheet. I'll also explain how to find the domain and range of each.

### 1. \( f(x) = -4x + 12 \)
This is a linear function, as it is in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

- **Domain:** A linear function has no restrictions, so the domain is all real numbers: \( (-\infty, \infty) \).
- **Range:** Since the function is linear and can take any real value as \( x \) changes, the range is also \( (-\infty, \infty) \).
  
To graph this, the slope \( m = -4 \) means that for each unit increase in \( x \), \( y \) decreases by 4. The y-intercept is 12, meaning the graph crosses the y-axis at \( (0, 12) \).

### 2. \( g(x) = -7 \)
This is a constant function.

- **Domain:** Since it is defined for all \( x \), the domain is \( (-\infty, \infty) \).
- **Range:** The range is just the constant value \( g(x) = -7 \), so the range is \( \{-7\} \).

The graph is a horizontal line at \( y = -7 \).

### 3. \( h(x) = -4x^2 - 16x + 1 \)
This is a quadratic function. To find the domain and range, we will need to analyze its vertex form. We can complete the square or use the vertex formula to find the vertex.

- **Domain:** A quadratic function is defined for all real values of \( x \), so the domain is \( (-\infty, \infty) \).
- **Range:** The parabola opens downward (since the coefficient of \( x^2 \) is negative). To find the range, we need the vertex. The vertex formula is:
  
  \[
  x = \frac{-b}{2a} = \frac{-(-16)}{2(-4)} = \frac{16}{-8} = -2
  \]
  
  Substituting \( x = -2 \) back into the function to find the corresponding \( y \)-value:
  
  \[
  h(-2) = -4(-2)^2 - 16(-2) + 1 = -4(4) + 32 + 1 = -16 + 32 + 1 = 17
  \]
  
  So the vertex is \( (-2, 17) \), and since the parabola opens downward, the range is \( (-\infty, 17] \).

### 4. \( r(x) = \sqrt{-2x + 6} \)
This is a square root function. The expression inside the square root must be non-negative for the function to be defined.

- **Domain:** We solve the inequality \( -2x + 6 \geq 0 \):
  
  \[
  -2x \geq -6 \quad \Rightarrow \quad x \leq 3
  \]
  
  So the domain is \( (-\infty, 3] \).

- **Range:** The square root function always produces non-negative outputs. Therefore, the range is \( [0, \infty) \).

### 5. \( f(x) = \sqrt{3x + 12} \)
This is also a square root function. Again, the expression inside the square root must be non-negative.

- **Domain:** Solve \( 3x + 12 \geq 0 \):
  
  \[
  3x \geq -12 \quad \Rightarrow \quad x \geq -4
  \]
  
  So the domain is \( [-4, \infty) \).

- **Range:** Since the square root function yields non-negative values, the range is \( [0, \infty) \).

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Let me know if you need any further clarifications, and here are five related questions you might want to explore further:

1. How do you find the vertex of a quadratic function algebraically?
2. What does the slope of a linear function represent geometrically?
3. Why do square root functions have restricted domains and ranges?
4. How would you graph a quadratic function by completing the square?
5. How can transformations (like shifts and stretches) affect the graph of a square root function?

**Tip:** Always check if the function has any domain restrictions (like square roots or divisions) before finding its range.

Graph and find the domain and range of the following functions: 1. f(x) = -4x + 12, 2. g(x) = -7, 3. h(x) = -4x^2 - 16x + 1, 4. r(x) = sqrt(-2x + 6), 5. f(x) = sqrt(3x + 12).

Explore how to find the domain and range of various functions, including linear, quadratic, and square root functions. The worksheet covers solving and graphing problems like f(x) = -4x + 12, g(x) = -7, and more, suitable for students in Grades 9-12.

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