what is the domain of the composite function f little circle g f(x) =$$ \sqrt{6-x};g\left(x\right)=\frac{1}{x-4} $$

To find the domain of the composite function $ (f \circ g)(x) = f(g(x)) $, where:

- $ f(x) = \sqrt{6 - x} $
- $ g(x) = \frac{1}{x - 4} $

we need to follow these steps:

### 1. Find the domain of $ g(x) $.
For $ g(x) = \frac{1}{x - 4} $, the denominator cannot be zero, so:
\[
x - 4 \neq 0 \quad \Rightarrow \quad x \neq 4
\]
Thus, the domain of $ g(x) $ is all real numbers except $ x = 4 $, or:
\[
\text{Domain of } g(x) = (-\infty, 4) \cup (4, \infty)
\]

### 2. Find the domain of $ f(g(x)) $.
Now, we apply $ g(x) $ inside $ f(x) $, so:
\[
f(g(x)) = f\left( \frac{1}{x - 4} \right) = \sqrt{6 - \frac{1}{x - 4}}
\]
For the square root to be defined, the expression inside the square root must be non-negative:
\[
6 - \frac{1}{x - 4} \geq 0
\]
Solve this inequality:
\[
6 \geq \frac{1}{x - 4}
\]
Multiply both sides by $ x - 4 $, keeping in mind that the direction of the inequality may change depending on whether $ x - 4 $ is positive or negative:

- **Case 1: $ x - 4 > 0 $ (i.e., $ x > 4 $)**

  In this case, multiply as usual:
  \[
  6(x - 4) \geq 1
  \]
  Simplifying:
  \[
  6x - 24 \geq 1 \quad \Rightarrow \quad 6x \geq 25 \quad \Rightarrow \quad x \geq \frac{25}{6}
  \]

- **Case 2: $ x - 4 < 0 $ (i.e., $ x < 4 $)**

  Here, the inequality flips when multiplying:
  \[
  6(x - 4) \leq 1
  \]
  Simplifying:
  \[
  6x - 24 \leq 1 \quad \Rightarrow \quad 6x \leq 25 \quad \Rightarrow \quad x \leq \frac{25}{6}
  \]

### 3. Combine the results.
From Case 1, we know $ x \geq \frac{25}{6} $, and from Case 2, $ x \leq \frac{25}{6} $. Therefore, the solution is that $ x = \frac{25}{6} $, and we must also exclude $ x = 4 $.

### Conclusion:
The domain of the composite function $ (f \circ g)(x) $ is $ \left\{ \frac{25}{6} \right\} $.

Would you like more details or have any further questions?

### Related questions:
1. How do you solve compound inequalities involving fractions?
2. What is the domain of the square root function in general?
3. How can you graphically represent composite functions?
4. How do you handle domains when composing two functions?
5. What are some common mistakes when finding domains of composite functions?

### Tip:
Always remember to check both the inner and outer functions' domains when dealing with composite functions.

What is the domain of the composite function f ∘ g where f(x) = √(6 - x) and g(x) = 1 / (x - 4)?

Discover how to find the domain of the composite function f ∘ g, where f(x) = √(6 - x) and g(x) = 1 / (x - 4). This step-by-step guide explores domain restrictions for square root and rational functions, including solving inequalities. Ideal for high school students in Grades 10-12.

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