f(x) = 1/x
g(x) = (x^2 - 2x + 3)/(x - 3)
Find the domain of the composite function f(g(x))

To find the domain of the composite function \( f(g(x)) \), we need to determine the values of \( x \) for which the composition \( f(g(x)) \) is defined.

### Steps:
1. **Domain of \( g(x) \):**
   \[
   g(x) = \frac{x^2 - 2x + 3}{x - 3}
   \]
   The function \( g(x) \) is undefined when the denominator is 0, i.e., when:
   \[
   x - 3 = 0 \implies x = 3
   \]
   So, the domain of \( g(x) \) is all real numbers except \( x = 3 \).

2. **Domain of \( f(x) \):**
   \[
   f(x) = \frac{1}{x}
   \]
   The function \( f(x) \) is undefined when \( x = 0 \), so for \( f(g(x)) \) to be defined, \( g(x) \neq 0 \).

3. **Solve \( g(x) = 0 \):**
   To ensure that \( f(g(x)) \) is defined, we need to find when \( g(x) = 0 \):
   \[
   \frac{x^2 - 2x + 3}{x - 3} = 0 \implies x^2 - 2x + 3 = 0
   \]
   The quadratic equation \( x^2 - 2x + 3 = 0 \) has no real solutions because its discriminant is negative:
   \[
   \Delta = (-2)^2 - 4(1)(3) = 4 - 12 = -8
   \]
   Since there are no real solutions to this equation, \( g(x) \) never equals 0.

### Conclusion:
The domain of \( f(g(x)) \) is the same as the domain of \( g(x) \), except for when \( g(x) = 0 \), which doesn't happen for any real \( x \). Therefore, the only restriction is that \( x \neq 3 \).

Thus, the domain of \( f(g(x)) \) is:
\[
x \in (-\infty, 3) \cup (3, \infty)
\]

Would you like a more detailed explanation of any step?

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Here are 5 related questions you might find useful:
1. How do you find the domain of the sum of two functions?
2. What happens to the domain if the composite function has a square root?
3. How do you find the range of a composite function?
4. What is the difference between a composite function and a product of functions?
5. Can the range of \( f(g(x)) \) affect its domain?

**Tip:** Always start by finding the domain of the inner function before considering the outer function when working with composite functions.

f(x) = 1/x, g(x) = (x^2 - 2x + 3)/(x - 3). Find the domain of the composite function f(g(x)).

Learn how to find the domain of the composite function f(g(x)), where f(x) = 1/x and g(x) = (x^2 - 2x + 3)/(x - 3). This problem involves determining the domain of a rational function and applying the quadratic equation discriminant. Suitable for students in Grades 9-11.

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