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.