Riley Hilton
12/25/2023 · Middle School
For \( f(x)=7 \) and \( g(x)=x^{2}+6 \), find the domains of \( f, g, \frac{1}{g} \), and \( \frac{y}{f} \)
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
To find the domains of the functions \( f(x) = 7 \) and \( g(x) = x^2 + 6 \), we will analyze each function individually.
1. **Domain of \( f(x) = 7 \)**:
- The function \( f(x) = 7 \) is a constant function. It is defined for all real numbers.
- **Domain of \( f \)**: \( (-\infty, \infty) \)
2. **Domain of \( g(x) = x^2 + 6 \)**:
- The function \( g(x) = x^2 + 6 \) is a polynomial function. Polynomial functions are also defined for all real numbers.
- **Domain of \( g \)**: \( (-\infty, \infty) \)
3. **Domain of \( \frac{1}{g} \)**:
- The function \( \frac{1}{g(x)} = \frac{1}{x^2 + 6} \) is defined as long as \( g(x) \neq 0 \).
- We need to check when \( g(x) = 0 \):
\[
x^2 + 6 = 0 \implies x^2 = -6
\]
This equation has no real solutions since the square of a real number cannot be negative. Therefore, \( g(x) \) is never zero.
- **Domain of \( \frac{1}{g} \)**: \( (-\infty, \infty) \)
4. **Domain of \( \frac{y}{f} \)**:
- The function \( \frac{y}{f(x)} = \frac{y}{7} \) is defined as long as \( f(x) \neq 0 \).
- Since \( f(x) = 7 \) is a constant and never zero, \( \frac{y}{f(x)} \) is defined for all real numbers \( y \).
- **Domain of \( \frac{y}{f} \)**: \( (-\infty, \infty) \)
In summary, the domains are:
- Domain of \( f \): \( (-\infty, \infty) \)
- Domain of \( g \): \( (-\infty, \infty) \)
- Domain of \( \frac{1}{g} \): \( (-\infty, \infty) \)
- Domain of \( \frac{y}{f} \): \( (-\infty, \infty) \)
Quick Answer
- Domain of \( f \): \( (-\infty, \infty) \)
- Domain of \( g \): \( (-\infty, \infty) \)
- Domain of \( \frac{1}{g} \): \( (-\infty, \infty) \)
- Domain of \( \frac{y}{f} \): \( (-\infty, \infty) \)
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