Use the given questions in operation on functions: f(x) = 3x - 2, g(x) = 5x + 2

Let's work through the operations on the given functions \( f(x) = 3x - 2 \) and \( g(x) = 5x + 2 \). Below are the basic operations we can perform with these functions:

### 1. **Addition of Functions:**
\[
(f + g)(x) = f(x) + g(x)
\]
\[
(f + g)(x) = (3x - 2) + (5x + 2) = 3x + 5x - 2 + 2 = 8x
\]
Thus, \( (f + g)(x) = 8x \).

### 2. **Subtraction of Functions:**
\[
(f - g)(x) = f(x) - g(x)
\]
\[
(f - g)(x) = (3x - 2) - (5x + 2) = 3x - 5x - 2 - 2 = -2x - 4
\]
Thus, \( (f - g)(x) = -2x - 4 \).

### 3. **Multiplication of Functions:**
\[
(f \cdot g)(x) = f(x) \cdot g(x)
\]
\[
(f \cdot g)(x) = (3x - 2)(5x + 2)
\]
Using the distributive property:
\[
(f \cdot g)(x) = 3x(5x + 2) - 2(5x + 2) = 15x^2 + 6x - 10x - 4 = 15x^2 - 4x - 4
\]
Thus, \( (f \cdot g)(x) = 15x^2 - 4x - 4 \).

### 4. **Division of Functions (where \( g(x) \neq 0 \)):**
\[
\left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)}
\]
\[
\left( \frac{f}{g} \right)(x) = \frac{3x - 2}{5x + 2}
\]
Thus, \( \left( \frac{f}{g} \right)(x) = \frac{3x - 2}{5x + 2} \), for \( x \neq -\frac{2}{5} \) to avoid division by zero.

### 5. **Composition of Functions:**
Let's compute \( f(g(x)) \) and \( g(f(x)) \).

#### a) **\( f(g(x)) \):**
\[
f(g(x)) = f(5x + 2) = 3(5x + 2) - 2 = 15x + 6 - 2 = 15x + 4
\]
Thus, \( f(g(x)) = 15x + 4 \).

#### b) **\( g(f(x)) \):**
\[
g(f(x)) = g(3x - 2) = 5(3x - 2) + 2 = 15x - 10 + 2 = 15x - 8
\]
Thus, \( g(f(x)) = 15x - 8 \).

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Would you like more details or explanations on any of these operations?

Here are 5 follow-up questions to expand on this:
1. What happens when we find the inverse of both \( f(x) \) and \( g(x) \)?
2. Can we find a real solution to \( f(x) = g(x) \)?
3. How does the composition \( f(f(x)) \) or \( g(g(x)) \) behave?
4. What is the domain and range of \( f(x) \) and \( g(x) \)?
5. Can we solve \( f(x)g(x) = 0 \)? If so, how?

**Tip:** When working with function composition, always apply the innermost function first.

Explore the operations on the functions f(x) = 3x - 2 and g(x) = 5x + 2. This page provides a detailed solution for addition, subtraction, multiplication, division, and composition of functions, suitable for high school students learning algebraic functions. Includes step-by-step explanations for key operations and formulas.

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