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The image shows a list of mathematical exercises involving various types of functions. Each exercise asks to **sketch the graph of the function and determine its domain and range**. Below is a breakdown of the types of functions presented in the image:

1. **Linear functions**:
   - Exercise 11: \( f(x) = 3x - 1 \)
   - Exercise 12: \( g(x) = -4 - x \)
   - Exercise 13: \( F(x) = 2x^2 \)

2. **Quadratic functions**:
   - Exercise 14: \( G(x) = x^2 + 2 \)
   - Exercise 16: \( f(x) = (x - 1)^2 \)
   - Exercise 15: \( g(x) = 5 - x^2 \)
   - Exercise 17: \( G(x) = \sqrt{x - 1} \)
   
3. **Square root functions**:
   - Exercise 18: \( F(x) = \sqrt{9 - x} \)
   - Exercise 19: \( f(x) = \sqrt{x^2 - 4} \)

4. **Absolute value functions**:
   - Exercise 23: \( h(x) = |x - 3| \)
   - Exercise 25: \( F(x) = |3x + 2| \)

5. **Rational functions**:
   - Exercise 27: \( H(x) = \frac{x^2 - 25}{x + 5} \)
   - Exercise 28: \( f(x) = \frac{2x^2 + 7x + 3}{x + 3} \)
   - Exercise 29: \( f(x) = \frac{x^2 - 4x + 3}{x - 1} \)

6. **Piecewise functions**:
   - Exercise 31: \( f(x) \) is defined piecewise with two intervals.
   - Exercise 32: \( g(x) \) is also defined piecewise.
   - Exercise 33-40 follow a similar piecewise pattern.

Would you like to explore any specific problem in detail, such as calculating the domain and range, or graphing any of these functions?

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Here are 5 related questions to explore:
1. How do you find the domain of a square root function like \( f(x) = \sqrt{x - 1} \)?
2. What is the general approach to finding the range of a rational function?
3. How do piecewise functions differ in terms of graphing compared to continuous functions?
4. What are the main properties of absolute value functions when graphing?
5. How do we determine vertical asymptotes for a rational function?

**Tip:** For rational functions, check for points where the denominator equals zero to identify domain restrictions and potential vertical asymptotes.

The image contains a list of mathematical exercises, asking to sketch the graph of functions and determine their domain and range.

A set of math exercises where students are tasked with graphing linear, quadratic, square root, absolute value, rational, and piecewise functions, and determining the domain and range. Suitable for high school students in Grades 9-12.

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