
answer an explain step by step this Function and relation in mathematics in the modern world  A. Given the function f(x)=2x² and g(x)= 3x+1, find: 1.f(x)+g(x) 2. f(x)-g(x) 3. f[g(x)] 4. f(x)² 5. f(x)³ - g(x)+g(x)²

Let's break down each part of the problem step by step. We are given two functions:

\[
f(x) = 2x^2 \quad \text{and} \quad g(x) = 3x + 1
\]

### 1. \( f(x) + g(x) \)

To find \( f(x) + g(x) \), simply add the two functions together:

\[
f(x) + g(x) = 2x^2 + (3x + 1)
\]

Combine the like terms:

\[
f(x) + g(x) = 2x^2 + 3x + 1
\]

### 2. \( f(x) - g(x) \)

Now, subtract \( g(x) \) from \( f(x) \):

\[
f(x) - g(x) = 2x^2 - (3x + 1)
\]

Distribute the minus sign:

\[
f(x) - g(x) = 2x^2 - 3x - 1
\]

### 3. \( f[g(x)] \)

To find \( f[g(x)] \), we need to substitute \( g(x) \) into \( f(x) \). Since \( f(x) = 2x^2 \), we substitute \( g(x) = 3x + 1 \) for \( x \) in \( f(x) \):

\[
f[g(x)] = 2(3x + 1)^2
\]

Now expand \( (3x + 1)^2 \):

\[
(3x + 1)^2 = 9x^2 + 6x + 1
\]

So:

\[
f[g(x)] = 2(9x^2 + 6x + 1)
\]

Distribute the 2:

\[
f[g(x)] = 18x^2 + 12x + 2
\]

### 4. \( f(x)^2 \)

Now, square \( f(x) \):

\[
f(x)^2 = (2x^2)^2
\]

Square the terms:

\[
f(x)^2 = 4x^4
\]

### 5. \( f(x)^3 - g(x) + g(x)^2 \)

This expression involves a combination of powers of \( f(x) \) and \( g(x) \). First, calculate each part:

\[
f(x)^3 = (2x^2)^3 = 8x^6
\]
\[
g(x) = 3x + 1
\]
\[
g(x)^2 = (3x + 1)^2 = 9x^2 + 6x + 1
\]

Now combine everything:

\[
f(x)^3 - g(x) + g(x)^2 = 8x^6 - (3x + 1) + (9x^2 + 6x + 1)
\]

Simplify the expression:

\[
f(x)^3 - g(x) + g(x)^2 = 8x^6 + 9x^2 + 3x
\]

### Final Answers:
1. \( f(x) + g(x) = 2x^2 + 3x + 1 \)
2. \( f(x) - g(x) = 2x^2 - 3x - 1 \)
3. \( f[g(x)] = 18x^2 + 12x + 2 \)
4. \( f(x)^2 = 4x^4 \)
5. \( f(x)^3 - g(x) + g(x)^2 = 8x^6 + 9x^2 + 3x \)

---

Would you like more details on any of these steps? Let me know if you have any questions!

Here are some related questions:
1. How would you evaluate \( f(x) + g(x) \) at a specific value of \( x \)?
2. What happens if you substitute different functions for \( f(x) \) and \( g(x) \) in these expressions?
3. How do function compositions like \( f[g(x)] \) relate to real-world problems?
4. Can you extend these operations to more than two functions, say \( f(x) \), \( g(x) \), and \( h(x) \)?
5. How would the result change if we used a more complex function, like \( f(x) = x^3 \)?

**Tip:** When working with function operations, always be mindful of the order of operations (like when distributing or composing functions) to avoid mistakes!

answer an explain step by step this Function and relation in mathematics in the modern world  A. Given the function f(x)=2x² and g(x)= 3x+1, find: 1.f(x)+g(x) 2. f(x)-g(x) 3. f[g(x)] 4. f(x)² 5. f(x)³ - g(x)+g(x)²

Learn how to perform various function operations, including addition, subtraction, composition, and exponentiation with functions f(x)=2x² and g(x)=3x+1. This guide provides detailed explanations for students in Grades 9-12, covering key algebraic concepts and operations on functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation