To solve a composite function problem involving $f(x)$ and $g(x)$, we first need to find either $f(g(x))$ or $g(f(x))$. Here's how we can compute each:

1. Definitions:

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

2. Calculating the Composite Functions:

$f(g(x))$:

1. Substitute $g(x)$ into $f$: $f(g(x)) = f(x^2 - 8x + 16)$
2. Using the formula for $f$, where $f(x) = x - 4$, replace $x$ with $g(x)$: $f(g(x)) = (x^2 - 8x + 16) - 4$
3. Simplify the expression: $f(g(x)) = x^2 - 8x + 12$

$g(f(x))$:

1. Substitute $f(x)$ into $g$: $g(f(x)) = g(x - 4)$
2. Using the formula for $g$, where $g(x) = x^2 - 8x + 16$, replace $x$ with $f(x)$: $g(f(x)) = (x - 4)^2 - 8(x - 4) + 16$
3. Expand the squared term and simplify: $g(f(x)) = (x^2 - 8x + 16) - 8(x - 4) + 16$ $g(f(x)) = x^2 - 8x + 16 - 8x + 32 + 16$ $g(f(x)) = x^2 - 16x + 64$

3. Summary:

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

Would you like a deeper breakdown of each step or have any questions?

Related Questions:

1. What happens if we evaluate $f(g(a))$ for a specific value of $a$?
2. Can we find the domain and range of $f(g(x))$?
3. What are the points of intersection between $f(g(x))$ and $g(f(x))$?
4. How does the behavior of $f$ and $g$ change the composite functions?
5. Can we find the inverse of $f(g(x))$?

Tip:

When finding composite functions, always substitute carefully and simplify step-by-step to avoid errors.