https://file.aiquickdraw.com/mathbot/file/1725456009717s55wswqa.jpg

To solve this problem, let's go through each part step by step, based on the functions provided in the image.

### Given Functions:
1. \( f(x) = x + 3 \)
2. \( g(x) = 2x - 5 \)
3. \( h(x) = 2x^2 + 3x + 2 \)
4. \( p(x) = 3x^2 - 3x + 7 \)
5. \( t(x) = x^2 - 9 \)

### Problems to Solve:

#### a. \((f + g)(x)\)
\[
(f + g)(x) = f(x) + g(x)
\]
\[
= (x + 3) + (2x - 5)
\]
\[
= x + 2x + 3 - 5
\]
\[
= 3x - 2
\]

#### b. \((t - g)(x)\)
\[
(t - g)(x) = t(x) - g(x)
\]
\[
= (x^2 - 9) - (2x - 5)
\]
\[
= x^2 - 9 - 2x + 5
\]
\[
= x^2 - 2x - 4
\]

#### c. \((h \cdot g)(x)\)
\[
(h \cdot g)(x) = h(x) \cdot g(x)
\]
\[
= (2x^2 + 3x + 2) \cdot (2x - 5)
\]

Expanding the multiplication:
\[
= 2x^2 \cdot 2x + 2x^2 \cdot (-5) + 3x \cdot 2x + 3x \cdot (-5) + 2 \cdot 2x + 2 \cdot (-5)
\]
\[
= 4x^3 - 10x^2 + 6x^2 - 15x + 4x - 10
\]
\[
= 4x^3 - 4x^2 - 11x - 10
\]

#### d. \((f + h)(x)\)
\[
(f + h)(x) = f(x) + h(x)
\]
\[
= (x + 3) + (2x^2 + 3x + 2)
\]
\[
= 2x^2 + 3x + x + 3 + 2
\]
\[
= 2x^2 + 4x + 5
\]

#### e. \(\frac{f}{p}(x)\)
\[
\frac{f}{p}(x) = \frac{f(x)}{p(x)}
\]
\[
= \frac{x + 3}{3x^2 - 3x + 7}
\]

This is the simplified form since the numerator and the denominator have no common factors.

#### f. \(\frac{g}{f}(x)\)
\[
\frac{g}{f}(x) = \frac{g(x)}{f(x)}
\]
\[
= \frac{2x - 5}{x + 3}
\]

This is the simplified form since the numerator and the denominator have no common factors.

### Summary of Answers:
1. \( (f + g)(x) = 3x - 2 \)
2. \( (t - g)(x) = x^2 - 2x - 4 \)
3. \( (h \cdot g)(x) = 4x^3 - 4x^2 - 11x - 10 \)
4. \( (f + h)(x) = 2x^2 + 4x + 5 \)
5. \( \frac{f}{p}(x) = \frac{x + 3}{3x^2 - 3x + 7} \)
6. \( \frac{g}{f}(x) = \frac{2x - 5}{x + 3} \)

Would you like more details or have any other questions?

### Further Questions:
1. What are the domain restrictions for \(\frac{f}{p}(x)\)?
2. How do you find the inverse of \(f(x)\)?
3. What is the composition of \(f(g(x))\)?
4. How do the graphs of \(f(x)\) and \(g(x)\) compare?
5. What are the critical points of \(h(x)\)?

**Tip:** When dealing with operations on functions, always ensure that the domain of the resulting function considers the domains of the individual functions involved.

Learn how to perform operations on functions with a detailed step-by-step solution, covering addition, subtraction, multiplication, and division of functions. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation