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Let's solve the given problem step by step.

### Given Functions:
- \( f(x) = 10x - 9 \)
- \( g(x) = \sqrt{x^2 - 6x + 10} \)

We need to find:

1. \( (f \circ g)(x) \), which means \( f(g(x)) \)
2. \( (f \circ g)(-4) \)

### 1. Finding \( (f \circ g)(x) \):
\( f(g(x)) \) means we substitute \( g(x) \) into \( f(x) \).

We know:
- \( f(x) = 10x - 9 \)
- \( g(x) = \sqrt{x^2 - 6x + 10} \)

So, \( f(g(x)) = 10 \cdot g(x) - 9 \).

Substitute \( g(x) \):
\[
f(g(x)) = 10 \cdot \sqrt{x^2 - 6x + 10} - 9
\]

Thus,
\[
(f \circ g)(x) = 10 \cdot \sqrt{x^2 - 6x + 10} - 9
\]

### 2. Finding \( (f \circ g)(-4) \):
We substitute \( x = -4 \) into \( g(x) \) first, and then find \( f(g(-4)) \).

First, calculate \( g(-4) \):
\[
g(-4) = \sqrt{(-4)^2 - 6(-4) + 10} = \sqrt{16 + 24 + 10} = \sqrt{50}
\]
So, \( g(-4) = \sqrt{50} \), which simplifies to \( g(-4) = 5\sqrt{2} \).

Now, substitute \( g(-4) = 5\sqrt{2} \) into \( f(x) \):
\[
f(g(-4)) = 10 \cdot 5\sqrt{2} - 9 = 50\sqrt{2} - 9
\]

Thus,
\[
(f \circ g)(-4) = 50\sqrt{2} - 9
\]

### Final Results:
1. \( (f \circ g)(x) = 10 \cdot \sqrt{x^2 - 6x + 10} - 9 \)
2. \( (f \circ g)(-4) = 50\sqrt{2} - 9 \)

Would you like more details on any part of this, or do you have further questions?

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Here are some related questions to expand on the topic:

1. How do you find the domain of a composite function?
2. What is the difference between \( f(g(x)) \) and \( g(f(x)) \)?
3. How do you handle the square root function when working with domains?
4. Can you calculate the range of \( (f \circ g)(x) \)?
5. How does function composition affect the graph of the resulting function?

**Tip:** Always check the domain of the inner function before composing two functions, especially when dealing with square roots or fractions!

Suppose f(x) = 10x - 9 and g(x) = sqrt(x^2 - 6x + 10). Find (f ∘ g)(x) and (f ∘ g)(-4).

Learn how to solve a function composition problem with f(x) = 10x - 9 and g(x) = sqrt(x^2 - 6x + 10). We break down the steps to find (f ∘ g)(x) and evaluate (f ∘ g)(-4). This problem covers important concepts in function composition and algebra.

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