We are given two functions:
$f(x) = 5x + 2$ and $g(x) = x^2 + 4$.

Let's evaluate each part step by step.

(a) $f(g(x))$

This is the composition of functions, where we plug $g(x)$ into $f(x)$.

1. First, recall $g(x) = x^2 + 4$.
2. Now, substitute $g(x)$ into $f(x)$: $f(g(x)) = f(x^2 + 4) = 5(x^2 + 4) + 2$
3. Simplify: $f(g(x)) = 5x^2 + 20 + 2 = 5x^2 + 22$

(b) $g(f(x))$

Now we will substitute $f(x)$ into $g(x)$.

1. Recall $f(x) = 5x + 2$.
2. Substitute $f(x)$ into $g(x)$: $g(f(x)) = g(5x + 2) = (5x + 2)^2 + 4$
3. Expand $(5x + 2)^2$: $(5x + 2)^2 = (5x)^2 + 2(5x)(2) + 2^2 = 25x^2 + 20x + 4$
4. Now, add 4 to the result: $g(f(x)) = 25x^2 + 20x + 4 + 4 = 25x^2 + 20x + 8$

(c) $(f ∘ g)(−2)$

This means we first evaluate $g(-2)$, then plug it into $f(x)$.

1. First, evaluate $g(-2)$: $g(-2) = (-2)^2 + 4 = 4 + 4 = 8$
2. Now, substitute $g(-2) = 8$ into $f(x)$: $f(8) = 5(8) + 2 = 40 + 2 = 42$ Thus, $(f ∘ g)(-2) = 42$.

(d) $(g ∘ f)(6)$

This means we first evaluate $f(6)$, then plug it into $g(x)$.

1. First, evaluate $f(6)$: $f(6) = 5(6) + 2 = 30 + 2 = 32$
2. Now, substitute $f(6) = 32$ into $g(x)$: $g(32) = 32^2 + 4 = 1024 + 4 = 1028$ Thus, $(g ∘ f)(6) = 1028$.

Final Answers:

(a) $f(g(x)) = 5x^2 + 22$
(b) $g(f(x)) = 25x^2 + 20x + 8$
(c) $(f ∘ g)(-2) = 42$
(d) $(g ∘ f)(6) = 1028$

Let me know if you'd like more details on any of these calculations or have questions!

Related questions:

1. What is the composition $f(f(x))$ for the given functions?
2. How would you graph $f(x) = 5x + 2$ and $g(x) = x^2 + 4$?
3. What is the domain and range of $f(g(x))$?
4. Can you find the inverse of $f(x)$? Does it exist?
5. How would the results change if $g(x) = x^3 + 4$?

Tip:

When composing functions, always substitute carefully and simplify step by step to avoid mistakes.