We are tasked with finding $f(g(x))$ (composition of $f$ and $g$) and $g(f(x))$ for the given functions in question 1:

Given:

$f(x) = x - 4$
$g(x) = 4 - x^2$

1. Finding $f(g(x))$:

To find $f(g(x))$, we substitute $g(x)$ into $f(x)$.

$f(g(x)) = f(4 - x^2) = (4 - x^2) - 4$ Simplifying: $f(g(x)) = 4 - x^2 - 4 = -x^2$ So, $f(g(x)) = -x^2$.

2. Finding $g(f(x))$:

To find $g(f(x))$, we substitute $f(x)$ into $g(x)$.

$g(f(x)) = g(x - 4) = 4 - (x - 4)^2$ Now expand $(x - 4)^2$: $(x - 4)^2 = x^2 - 8x + 16$ Thus: $g(f(x)) = 4 - (x^2 - 8x + 16) = 4 - x^2 + 8x - 16$ Simplifying: $g(f(x)) = -x^2 + 8x - 12$ So, $g(f(x)) = -x^2 + 8x - 12$.

Summary:

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

Would you like further details or explanations?
Here are 5 related questions for you to explore:

1. How do we interpret the composition of two functions geometrically?
2. What is the domain of $f(g(x))$ when the inner function has a restricted domain?
3. Can you derive the inverse of the function $g(x) = 4 - x^2$?
4. How would you verify if two composed functions are inverses of each other?
5. What happens if the compositions $f(g(x))$ and $g(f(x))$ are the same function?

Tip: Always ensure that the domain of the composed function $f(g(x))$ is valid by considering the domain restrictions of both functions.