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Let's analyze the problem presented in the image.

### Problem Breakdown:
1. **Functions Given:**
   - \( f(x) = \sqrt{x - 3} \)
   - \( g(x) = x^2 + 2x \)

2. **Function Composition:**
   - The function \( f \circ g(x) \) is defined for all \( x \in \mathbb{R} \), except for the interval \( ]a, b[ \).

#### Part (a) Calculate the values of \( a \) and \( b \).

- To determine the values of \( a \) and \( b \), we need to consider where the function \( f(g(x)) \) is not defined.

  - The function \( f(x) \) involves a square root, which means \( f(x) \) is only defined when \( x - 3 \geq 0 \), i.e., \( x \geq 3 \).
  - Therefore, for \( f(g(x)) \) to be defined, \( g(x) \) must satisfy \( g(x) \geq 3 \).

  Let's solve \( g(x) \geq 3 \):

  \[
  g(x) = x^2 + 2x
  \]
  \[
  x^2 + 2x \geq 3
  \]
  \[
  x^2 + 2x - 3 \geq 0
  \]

  Now, factorize the quadratic expression:
  \[
  (x + 3)(x - 1) \geq 0
  \]

  The critical points are \( x = -3 \) and \( x = 1 \). The quadratic expression changes signs at these points.

  Analyzing the intervals:
  - \( x \in (-\infty, -3] \) and \( x \in [1, \infty) \) make the expression non-negative.
  - \( x \in (-3, 1) \) makes the expression negative.

  Therefore, \( f(g(x)) \) is **not defined** on the interval \( (-3, 1) \). So, \( a = -3 \) and \( b = 1 \).

#### Part (b) Find the range of \( f \circ g \).

- The range of \( f(g(x)) \) is the set of all possible values \( f(g(x)) \) can take when \( x \) lies in the domains where \( f(g(x)) \) is defined, i.e., \( x \in (-\infty, -3] \cup [1, \infty) \).

- For \( x \in (-\infty, -3] \cup [1, \infty) \), \( g(x) \geq 3 \). So, \( f(g(x)) = \sqrt{g(x) - 3} \).

- As \( g(x) \) can take values from \( 3 \) to infinity within the specified domain, \( f(g(x)) \) will take values from \( 0 \) to infinity.

  Hence, the range of \( f \circ g(x) \) is \( [0, \infty) \).

### Summary:
(a) The values of \( a \) and \( b \) are \( a = -3 \) and \( b = 1 \).

(b) The range of \( f \circ g(x) \) is \( [0, \infty) \).

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Would you like more details on any of the steps? Here are five questions you might explore further:

1. How do you find the domain of a composite function?
2. What are the properties of square root functions that influence their domains and ranges?
3. How does factoring a quadratic equation help in finding the critical points?
4. What would happen to the domain and range if \( f(x) \) or \( g(x) \) were different functions?
5. Can you explain why the range of \( f \circ g \) starts at 0 and extends to infinity?

**Tip:** Always analyze the domain of individual functions before dealing with their compositions, as this determines where the composition is defined.

The functions f and g are defined as f(x) = √(x - 3) and g(x) = x² + 2x. The function f ∘ g(x) is defined for all x ∈ ℝ, except for the interval ]a, b[. (a) Calculate the values of a and b. (b) Find the range of f ∘ g.

Explore the domain and range of composite functions with the problem where f(x) = √(x - 3) and g(x) = x² + 2x. Learn how to solve for the interval ]a, b[ and find the range of the composite function. Suitable for students in Grades 10-12.

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