The functions fand g are defined as follows. f(x) = g(x) = X x +4 x-5 x²+x-30 For each function, find the domain. Write each answer as an interval or union of intervals. Domain off: Domain of g :
The functions fand g are defined as follows. f(x) = g(x) = X x +4 x-5 x²+x-30 For each function, find the domain. Write each answer as an interval or union of intervals. Domain off: Domain of g :
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![### Functions and Their Domains
The functions \( f \) and \( g \) are defined as follows:
\[ f(x) = \frac{x}{x^2 + 4} \]
\[ g(x) = \frac{x - 5}{x^2 + x - 30} \]
#### Problem Statement
For each function, find the domain. Write each answer as an interval or union of intervals.
#### Input Boxes for Answers
- **Domain of \( f \)**: \(\underline{\ \ \ \ \ }\)
- **Domain of \( g \)**: \(\underline{\ \ \ \ \ }\)
#### Interval and Infinity Symbols
The exercise includes the following symbols to express intervals:
- **Open interval**: \(( \ \ )\)
- **Closed interval**: \([ \ \ ])\)
- **Open from one side, closed from the other**: \(( \ ] \) or \([ \ )\)
- **Infinity symbols**: \(\infty\) and \(-\infty\)
- **Union**: \(\cup\)
- **Empty set**: \(\emptyset\)
#### Instructions
- Determine the domain of \( f \) and \( g \), and write your answer using the appropriate interval notation.
This problem can usually be found in introductory calculus or algebra courses, focusing on understanding the domain of rational functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa6b5c4dd-afec-429d-a37f-d528e4b9e4fe%2Fe43c28df-3bb7-4522-a3e2-e0a23dd76c64%2F4n8y2rj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Functions and Their Domains
The functions \( f \) and \( g \) are defined as follows:
\[ f(x) = \frac{x}{x^2 + 4} \]
\[ g(x) = \frac{x - 5}{x^2 + x - 30} \]
#### Problem Statement
For each function, find the domain. Write each answer as an interval or union of intervals.
#### Input Boxes for Answers
- **Domain of \( f \)**: \(\underline{\ \ \ \ \ }\)
- **Domain of \( g \)**: \(\underline{\ \ \ \ \ }\)
#### Interval and Infinity Symbols
The exercise includes the following symbols to express intervals:
- **Open interval**: \(( \ \ )\)
- **Closed interval**: \([ \ \ ])\)
- **Open from one side, closed from the other**: \(( \ ] \) or \([ \ )\)
- **Infinity symbols**: \(\infty\) and \(-\infty\)
- **Union**: \(\cup\)
- **Empty set**: \(\emptyset\)
#### Instructions
- Determine the domain of \( f \) and \( g \), and write your answer using the appropriate interval notation.
This problem can usually be found in introductory calculus or algebra courses, focusing on understanding the domain of rational functions.
![### Example Problem: Finding the Slope of a Line
**Problem Statement:**
Find the slope of the line passing through the points \((2, 5)\) and \((8, -4)\).
**Solution Explanation:**
To find the slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the slope formula:
\[ \text{Slope} (m) = \frac{y_2 - y_1}{x_2 - x_1} \]
Given points:
- \((x_1, y_1) = (2, 5)\)
- \((x_2, y_2) = (8, -4)\)
Substitute the values into the formula:
\[ m = \frac{-4 - 5}{8 - 2} = \frac{-9}{6} = -\frac{3}{2} \]
So, the slope of the line passing through the points \((2, 5)\) and \((8, -4)\) is \(-\frac{3}{2}\).
**Interactive Feature:**
The problem includes an interactive component where you can enter the slope of the line in a text box. Below the input box, there are four options/buttons labeled "X", "Undo", and "Help" for further assistance or to reset the input if needed.
In this case, the correct answer to input in the box is \(-\frac{3}{2}\).
**Visual Representation:**
While this problem does not include graphs or diagrams, understanding the concept of slope can often be aided by drawing the line through the given points on a coordinate plane and visually confirming the rise over run calculation.
Feel free to click on the help button for more guidance on solving slope problems!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa6b5c4dd-afec-429d-a37f-d528e4b9e4fe%2Fe43c28df-3bb7-4522-a3e2-e0a23dd76c64%2Fk3elqh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Example Problem: Finding the Slope of a Line
**Problem Statement:**
Find the slope of the line passing through the points \((2, 5)\) and \((8, -4)\).
**Solution Explanation:**
To find the slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the slope formula:
\[ \text{Slope} (m) = \frac{y_2 - y_1}{x_2 - x_1} \]
Given points:
- \((x_1, y_1) = (2, 5)\)
- \((x_2, y_2) = (8, -4)\)
Substitute the values into the formula:
\[ m = \frac{-4 - 5}{8 - 2} = \frac{-9}{6} = -\frac{3}{2} \]
So, the slope of the line passing through the points \((2, 5)\) and \((8, -4)\) is \(-\frac{3}{2}\).
**Interactive Feature:**
The problem includes an interactive component where you can enter the slope of the line in a text box. Below the input box, there are four options/buttons labeled "X", "Undo", and "Help" for further assistance or to reset the input if needed.
In this case, the correct answer to input in the box is \(-\frac{3}{2}\).
**Visual Representation:**
While this problem does not include graphs or diagrams, understanding the concept of slope can often be aided by drawing the line through the given points on a coordinate plane and visually confirming the rise over run calculation.
Feel free to click on the help button for more guidance on solving slope problems!
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