1. Given: f(x) = 5x + 2 and g(x) = x^ 2 + 4

Evaluate each of the following:
a. f(g(x))
b. g(f(x))
c. (f ∘ g)(−2)
d. (g ∘ f)(6)

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.

Given: f(x) = 5x + 2 and g(x) = x^2 + 4

Evaluate each of the following:
a. f(g(x))
b. g(f(x))
c. (f ∘ g)(−2)
d. (g ∘ f)(6)

Explore how to evaluate the composition of functions for f(x) = 5x + 2 and g(x) = x^2 + 4. This problem covers key algebraic concepts like function composition, polynomial expansion, and quadratic equations. Learn the step-by-step solution for questions such as f(g(x)), g(f(x)), and specific evaluations at x = -2 and x = 6.

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