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 $[0, \infty)$.

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.