Willis Tucker
07/24/2024 · Middle School
Omework The functions \( f \) and \( g \) are defined as \( f(x)=4 x-3 \) and \( g(x)=-9 x^{2} \) a) Find the domain of \( f, g, f+g, f-g, f g, f f, \frac{f}{g} \), and \( \frac{g}{f} \) b) Find \( (f+g)(x),(f-g)(x),(f g)(x),(f f)(x),\left(\frac{f}{g}\right)(x) \), and \( \left(\frac{g}{f}\right)(x) \). b) \( (f+g)(x)=-9 x^{2}+4 x-3 \) (Simplify your answer.) ( \( f-g)(x)=9 x^{2}+4 x-3 \) (Simplify your answer.) (fg) \( (x)=-36 x^{3}+27 x^{2} \) (Simplify your answer.) (f) \( (x)=\square( \) Simplify your answer.)
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
Let's solve the problem step by step.
### Part a: Finding the Domain
1. **Domain of \( f(x) = 4x - 3 \)**:
- This is a linear function, which is defined for all real numbers.
- **Domain of \( f \)**: \( \mathbb{R} \)
2. **Domain of \( g(x) = -9x^2 \)**:
- This is also a polynomial function, which is defined for all real numbers.
- **Domain of \( g \)**: \( \mathbb{R} \)
3. **Domain of \( f + g \)**:
- Since both \( f \) and \( g \) are defined for all real numbers, their sum is also defined for all real numbers.
- **Domain of \( f + g \)**: \( \mathbb{R} \)
4. **Domain of \( f - g \)**:
- Similar to the sum, the difference is also defined for all real numbers.
- **Domain of \( f - g \)**: \( \mathbb{R} \)
5. **Domain of \( fg \)**:
- The product of two functions that are defined for all real numbers is also defined for all real numbers.
- **Domain of \( fg \)**: \( \mathbb{R} \)
6. **Domain of \( \frac{f}{g} \)**:
- This function is defined wherever \( g(x) \neq 0 \).
- Since \( g(x) = -9x^2 \), it is zero when \( x = 0 \).
- **Domain of \( \frac{f}{g} \)**: \( \mathbb{R} \setminus \{0\} \)
7. **Domain of \( \frac{g}{f} \)**:
- This function is defined wherever \( f(x) \neq 0 \).
- Since \( f(x) = 4x - 3 \), it is zero when \( x = \frac{3}{4} \).
- **Domain of \( \frac{g}{f} \)**: \( \mathbb{R} \setminus \left\{ \frac{3}{4} \right\} \)
### Summary of Domains:
- \( \text{Domain of } f: \mathbb{R} \)
- \( \text{Domain of } g: \mathbb{R} \)
- \( \text{Domain of } f + g: \mathbb{R} \)
- \( \text{Domain of } f - g: \mathbb{R} \)
- \( \text{Domain of } fg: \mathbb{R} \)
- \( \text{Domain of } \frac{f}{g}: \mathbb{R} \setminus \{0\} \)
- \( \text{Domain of } \frac{g}{f}: \mathbb{R} \setminus \left\{ \frac{3}{4} \right\} \)
### Part b: Finding the Functions
1. **Finding \( (f + g)(x) \)**:
\[
(f + g)(x) = f(x) + g(x) = (4x - 3) + (-9x^2) = -9x^2 + 4x - 3
\]
2. **Finding \( (f - g)(x) \)**:
\[
(f - g)(x) = f(x) - g(x) = (4x - 3) - (-9x^2) = 4x - 3 + 9x^2 = 9x^2 + 4x - 3
\]
3. **Finding \( (fg)(x) \)**:
\[
(fg)(x) = f(x) \cdot g(x) = (4x - 3)(-9x^2) = -36x^3 + 27x^2
\]
4. **Finding \( (ff)(x) \)**:
\[
(ff)(x) = f(f(x)) = f(4x - 3) = 4(4x - 3) - 3 = 16x - 12 - 3 = 16x - 15
\]
5. **Finding \( \left(\frac{f}{g}\right)(x) \)**:
\[
\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{4x
Quick Answer
- Part a: Domains
- \( f \): \( \mathbb{R} \)
- \( g \): \( \mathbb{R} \)
- \( f + g \): \( \mathbb{R} \)
- \( f - g \): \( \mathbb{R} \)
- \( fg \): \( \mathbb{R} \)
- \( \frac{f}{g} \): \( \mathbb{R} \setminus \{0\} \)
- \( \frac{g}{f} \): \( \mathbb{R} \setminus \left\{ \frac{3}{4} \right\} \)
- Part b: Functions
- \( (f + g)(x) \): \( -9x^2 + 4x - 3 \)
- \( (f - g)(x) \): \( 9x^2 + 4x - 3 \)
- \( (fg)(x) \): \( -36x^3 + 27x^2 \)
- \( (ff)(x) \): \( 16x - 15 \)
Answered by UpStudy AI and reviewed by a Professional Tutor
UpStudy ThothAI
Self-Developed and Ever-Improving
Thoth AI product is constantly being upgraded and optimized.
Covers All Major Subjects
Capable of handling homework in math, chemistry, biology, physics, and more.
Instant and Accurate
Provides immediate and precise solutions and guidance.
Try Now
Ask Tutors
Ask AI
10x
Fastest way to Get Answers & Solutions
By text
Enter your question here…
By image
Re-Upload
Submit