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Let's solve the problem step by step. The functions are given as:

\[
f(x) = 3x + 5 \quad \text{and} \quad g(x) = 5x^2 + 5x.
\]

We need to calculate the compositions of functions for each part.

### 1. \( f \circ g \) (f of g)

This means \( f(g(x)) \). We substitute \( g(x) = 5x^2 + 5x \) into \( f(x) \):

\[
f(g(x)) = 3(5x^2 + 5x) + 5 = 15x^2 + 15x + 5.
\]

### 2. \( g \circ g \) (g of g)

This means \( g(g(x)) \). We substitute \( g(x) = 5x^2 + 5x \) into \( g(x) \):

\[
g(g(x)) = 5(5x^2 + 5x)^2 + 5(5x^2 + 5x).
\]
We first need to expand \( (5x^2 + 5x)^2 \):

\[
(5x^2 + 5x)^2 = 25x^4 + 50x^3 + 25x^2.
\]
Now substitute this into \( g(g(x)) \):

\[
g(g(x)) = 5(25x^4 + 50x^3 + 25x^2) + 5(5x^2 + 5x) = 125x^4 + 250x^3 + 125x^2 + 25x^2 + 25x = 125x^4 + 250x^3 + 150x^2 + 25x.
\]

### 3. \( g \circ f \) (g of f)

This means \( g(f(x)) \). We substitute \( f(x) = 3x + 5 \) into \( g(x) \):

\[
g(f(x)) = 5(3x + 5)^2 + 5(3x + 5).
\]
First, expand \( (3x + 5)^2 \):

\[
(3x + 5)^2 = 9x^2 + 30x + 25.
\]
Now substitute this into \( g(f(x)) \):

\[
g(f(x)) = 5(9x^2 + 30x + 25) + 5(3x + 5) = 45x^2 + 150x + 125 + 15x + 25 = 45x^2 + 165x + 150.
\]

### 4. \( f \circ f \) (f of f)

This means \( f(f(x)) \). We substitute \( f(x) = 3x + 5 \) into \( f(x) \):

\[
f(f(x)) = 3(3x + 5) + 5 = 9x + 15 + 5 = 9x + 20.
\]

### Final answers:

- \( f \circ g = 15x^2 + 15x + 5 \)
- \( g \circ g = 125x^4 + 250x^3 + 150x^2 + 25x \)
- \( g \circ f = 45x^2 + 165x + 150 \)
- \( f \circ f = 9x + 20 \)

Let me know if you need more details or clarifications.

Here are 5 related questions:
1. What is the general definition of function composition?
2. How can we graph the composition of two functions?
3. What are the domain restrictions for \( f \circ g(x) \)?
4. How does composition differ from multiplication of functions?
5. How can we reverse a composition of functions (inverse functions)?

**Tip:** When composing functions, always pay attention to the domain of the inner function, as it affects the domain of the composite function.

Suppose f(x) = 3x + 5 and g(x) = 5x^2 + 5x. Find the compositions for the following: 1) f ∘ g 2) g ∘ g 3) g ∘ f 4) f ∘ f.

Learn how to solve function compositions involving f(x) = 3x + 5 and g(x) = 5x^2 + 5x. This problem covers key algebraic concepts, including function composition and polynomial functions. Suitable for high school students.

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