The image contains a mathematical problem related to function composition. Let's break down and solve each part:

Given:

• $f(x) = x^2 + 4$
• $g(x) = x^2 - 3$

You are asked to find:

a. $(f \circ g)(x)$

This means $f(g(x))$.

b. $(g \circ f)(x)$

This means $g(f(x))$.

c. $(f \circ g)(4)$

This means $f(g(4))$.

d. $(g \circ f)(4)$

This means $g(f(4))$.

Let's calculate each one step by step.

a. $(f \circ g)(x)$

We substitute $g(x)$ into $f(x)$: $f(g(x)) = f(x^2 - 3) = (x^2 - 3)^2 + 4$ Now, simplify: $(x^2 - 3)^2 = x^4 - 6x^2 + 9$ So: $f(g(x)) = x^4 - 6x^2 + 9 + 4 = x^4 - 6x^2 + 13$

b. $(g \circ f)(x)$

We substitute $f(x)$ into $g(x)$: $g(f(x)) = g(x^2 + 4) = (x^2 + 4)^2 - 3$ Now, simplify: $(x^2 + 4)^2 = x^4 + 8x^2 + 16$ So: $g(f(x)) = x^4 + 8x^2 + 16 - 3 = x^4 + 8x^2 + 13$

c. $(f \circ g)(4)$

First, calculate $g(4)$: $g(4) = 4^2 - 3 = 16 - 3 = 13$ Now, substitute into $f$: $f(13) = 13^2 + 4 = 169 + 4 = 173$

d. $(g \circ f)(4)$

First, calculate $f(4)$: $f(4) = 4^2 + 4 = 16 + 4 = 20$ Now, substitute into $g$: $g(20) = 20^2 - 3 = 400 - 3 = 397$

Final Answers:

• $(f \circ g)(x) = x^4 - 6x^2 + 13$
• $(g \circ f)(x) = x^4 + 8x^2 + 13$
• $(f \circ g)(4) = 173$
• $(g \circ f)(4) = 397$

Would you like more details on any part of the solution, or do you have any questions?

Here are 5 related questions to extend this topic:

1. How do you prove the general composition of two functions $f(g(x))$?
2. What are some practical applications of function composition?
3. Can you express $f \circ g$ and $g \circ f$ graphically?
4. How do inverses of functions relate to function composition?
5. How does the order of function composition affect the result?

Tip: Always remember that the order of composition matters—$f(g(x))$ is generally different from $g(f(x))$.