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Let's break down the given problem step by step. You are tasked with finding the following compositions of functions:

1. \((f \circ g)(x)\)
2. \((g \circ f)(x)\)
3. \((g \circ f)(-5)\)

### Given Functions:
- \( f(x) = x^2 - 8x + 12 \)
- \( g(x) = x - 5 \)

### Step 1: Find \((f \circ g)(x)\)
This represents the composition of \(f(x)\) and \(g(x)\), meaning you substitute \(g(x)\) into \(f(x)\).

\[
(f \circ g)(x) = f(g(x)) = f(x - 5)
\]
Now, substitute \(x - 5\) into \(f(x)\):

\[
f(x - 5) = (x - 5)^2 - 8(x - 5) + 12
\]

Expand each term:
\[
(x - 5)^2 = x^2 - 10x + 25
\]
\[
-8(x - 5) = -8x + 40
\]

Now substitute everything back:
\[
f(x - 5) = (x^2 - 10x + 25) + (-8x + 40) + 12
\]
Simplify:
\[
f(x - 5) = x^2 - 18x + 77
\]
Thus, \((f \circ g)(x) = x^2 - 18x + 77\).

### Step 2: Find \((g \circ f)(x)\)
This represents the composition of \(g(x)\) and \(f(x)\), meaning you substitute \(f(x)\) into \(g(x)\).

\[
(g \circ f)(x) = g(f(x)) = g(x^2 - 8x + 12)
\]

Now substitute \(x^2 - 8x + 12\) into \(g(x)\):
\[
g(x^2 - 8x + 12) = (x^2 - 8x + 12) - 5
\]
Simplify:
\[
g(x^2 - 8x + 12) = x^2 - 8x + 7
\]
Thus, \((g \circ f)(x) = x^2 - 8x + 7\).

### Step 3: Find \((g \circ f)(-5)\)
This is the evaluation of \((g \circ f)(x)\) at \(x = -5\). From Step 2, we know that:

\[
(g \circ f)(x) = x^2 - 8x + 7
\]

Substitute \(x = -5\):
\[
(g \circ f)(-5) = (-5)^2 - 8(-5) + 7
\]
\[
= 25 + 40 + 7 = 72
\]

Thus, \((g \circ f)(-5) = 72\).

### Final Answers:
1. \((f \circ g)(x) = x^2 - 18x + 77\)
2. \((g \circ f)(x) = x^2 - 8x + 7\)
3. \((g \circ f)(-5) = 72\)

Would you like more details on any step, or do you have any questions?

### Related Questions:
1. What is the domain of \((f \circ g)(x)\) and \((g \circ f)(x)\)?
2. How would you find the inverse of \(f(x)\) or \(g(x)\)?
3. What does the composition of functions represent in real-world contexts?
4. Can a function \(f(x)\) ever equal its own inverse?
5. How do compositions of functions affect the range and domain?

### Tip:
When composing functions, always check the domain of the inner function to ensure the composition is valid.

Given f(x) = x^2 - 8x + 12 and g(x) = x - 5, find (f ∘ g)(x), (g ∘ f)(x), and (g ∘ f)(-5).

Learn how to solve function compositions with quadratic and linear functions. This problem includes step-by-step solutions for (f ∘ g)(x), (g ∘ f)(x), and evaluating (g ∘ f)(-5). Suitable for students in Grades 10-12.

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