To find the composition of two functions, we need to determine $(f \circ g)(x)$ and $(g \circ f)(x)$, which represent $f(g(x))$ and $g(f(x))$, respectively.

1. $(f \circ g)(x) = f(g(x))$

Given:

• $f(x) = x - 4$
• $g(x) = x^2 - 8x + 16$

To compute $f(g(x))$, substitute $g(x)$ into $f(x)$:

$f(g(x)) = f(x^2 - 8x + 16) = (x^2 - 8x + 16) - 4$

Simplifying:

$f(g(x)) = x^2 - 8x + 12$

2. $(g \circ f)(x) = g(f(x))$

Now, let's compute $g(f(x))$ by substituting $f(x)$ into $g(x)$:

$g(f(x)) = g(x - 4) = (x - 4)^2 - 8(x - 4) + 16$

First, expand $(x - 4)^2$:

$(x - 4)^2 = x^2 - 8x + 16$

Now expand $-8(x - 4)$:

$-8(x - 4) = -8x + 32$

Substitute these into the equation:

$g(f(x)) = (x^2 - 8x + 16) + (-8x + 32) + 16$

Simplify:

$g(f(x)) = x^2 - 16x + 64$

Final Results:

• $f(g(x)) = x^2 - 8x + 12$
• $g(f(x)) = x^2 - 16x + 64$

Would you like further details or have any questions?

Here are 5 related questions you might explore:

1. What happens to the compositions if we change the functions slightly?
2. How do you graph $f(g(x))$ and $g(f(x))$?
3. Can we find the domain of $f(g(x))$ and $g(f(x))$?
4. What if we compose more than two functions, like $f(g(h(x)))$?
5. How does composition differ from addition and multiplication of functions?

Tip: When composing functions, always remember to substitute the entire second function into the first one, being careful with parentheses and order of operations!