To find the domain of the composite function $f \circ g$, we need to understand the domain restrictions for both functions $f(x)$ and $g(x)$ individually and then for the composition.

Given:

• $f(x) = \frac{2}{x - 7}$
• $g(x) = \sqrt{x - 2}$

1. Domain of $g(x)$:
   The square root function $g(x) = \sqrt{x - 2}$ requires that $x - 2 \geq 0$.
   Solving this inequality: $x \geq 2$ So, the domain of $g(x)$ is $x \geq 2$.

2. Domain of $f(g(x))$:
   For the composite function $f(g(x))$, we need $g(x) \neq 7$ because of the denominator in $f(x)$.
   Setting $g(x) = 7$: $\sqrt{x - 2} = 7$ Squaring both sides: $x - 2 = 49 \Rightarrow x = 51$ Therefore, $x \neq 51$ to avoid division by zero in $f(g(x))$.

3. Combined Domain:
   Combining these restrictions, we have:

   • $x \geq 2$
   • $x \neq 51$

   The correct answer is: $\{ x \mid x \geq 2, x \neq 51 \}$

Answer

The correct option is:

• $\{ x \mid x \geq 2, x \neq 51 \}$

Would you like further details on the solution process?

Here are some related questions you might explore:

1. How do we determine the domain of a function involving a square root?
2. Why does $x = 51$ need to be excluded in this composite function?
3. How do restrictions in the inner function $g(x)$ affect the domain of $f(g(x))$?
4. What other types of functions require special attention for domain restrictions?
5. How would the answer change if $f(x)$ had a different form?

Tip: When determining the domain of composite functions, always start with the inner function and work outward, applying any additional restrictions introduced by the outer function.