Westonci.ca is the ultimate Q&A platform, offering detailed and reliable answers from a knowledgeable community. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
To determine how the mathematical domain and reasonable domain compare for the function [tex]\( f(x) = -0.05x^2 + 5x + 5.2 \)[/tex], consider the following steps:
### Step 1: Determine the Mathematical Domain
The mathematical domain of a function specifies all possible input values (x-values) for which the function is defined.
- Quadratic Function: For the function [tex]\( f(x) = -0.05x^2 + 5x + 5.2 \)[/tex], it is a quadratic function.
- General Quadratic Domain: For any quadratic function [tex]\( ax^2 + bx + c \)[/tex], the mathematical domain includes all real numbers, because quadratic functions are defined for all x-values.
Thus, the mathematical domain is [tex]\( -\infty < x < \infty \)[/tex].
### Step 2: Determine the Reasonable Domain
The reasonable domain is constrained by the context in which the function is used and represents the range of x-values that are meaningful in that particular application.
- Context: In this scenario, [tex]\( x \)[/tex] represents the distance from the goalie in a soccer game, and [tex]\( f(x) \)[/tex] represents the height of the ball.
- Reasonable Constraints:
- Starting Point: The ball is kicked from the goalie's position, so [tex]\( x \)[/tex] starts from 0.
- End Point: The reasonable end point will depend on the context, indicating how far the ball can realistically travel and still be meaningful in the game. For practical purposes, in a soccer game, we need to find an upper limit to [tex]\( x \)[/tex] that makes sense.
### Step 3: Compare Given Options for Reasonable Domain
- Option 1: Reasonable domain: [tex]\( 0 \leq x \leq 25 \)[/tex]
- Option 2: Reasonable domain: [tex]\( 0 \leq x \leq 52 \)[/tex]
- Option 3: Reasonable domain: [tex]\( 0 \leq x \leq 25 \)[/tex]
- Option 4: Reasonable domain: [tex]\( 0 \leq x \leq 5 \)[/tex]
### Analysis of Options
We need to find an option where the mathematical domain is [tex]\( -\infty < x < \infty \)[/tex] and the reasonable domain makes practical sense.
Upon reviewing all given options:
- Mathematical Domain: We see that options 3 and 4 both state [tex]\( -\infty < x < \infty \)[/tex], which is correct for the mathematical domain.
- Reasonable for [tex]\( x \)[/tex]:
- Option 3 suggests a reasonable domain [tex]\( 0 \leq x \leq 25 \)[/tex], which is appropriate for a realistic distance in a soccer field scenario.
- Option 4 suggests [tex]\( 0 \leq x \leq 5 \)[/tex], which might be too short for practical purposes.
Given the context and practical applications, Option 3 is the most appropriate.
### Conclusion
Based on the above analysis, the mathematical domain and reasonable domain compare as follows:
- Mathematical Domain: [tex]\( -\infty < x < \infty \)[/tex]
- Reasonable Domain: [tex]\( 0 \leq x \leq 25 \)[/tex]
Thus, the correct answer is:
- Mathematical Domain: [tex]\( -\infty < x < \infty \)[/tex]
- Reasonable Domain: [tex]\( 0 \leq x \leq 25 \)[/tex]
### Step 1: Determine the Mathematical Domain
The mathematical domain of a function specifies all possible input values (x-values) for which the function is defined.
- Quadratic Function: For the function [tex]\( f(x) = -0.05x^2 + 5x + 5.2 \)[/tex], it is a quadratic function.
- General Quadratic Domain: For any quadratic function [tex]\( ax^2 + bx + c \)[/tex], the mathematical domain includes all real numbers, because quadratic functions are defined for all x-values.
Thus, the mathematical domain is [tex]\( -\infty < x < \infty \)[/tex].
### Step 2: Determine the Reasonable Domain
The reasonable domain is constrained by the context in which the function is used and represents the range of x-values that are meaningful in that particular application.
- Context: In this scenario, [tex]\( x \)[/tex] represents the distance from the goalie in a soccer game, and [tex]\( f(x) \)[/tex] represents the height of the ball.
- Reasonable Constraints:
- Starting Point: The ball is kicked from the goalie's position, so [tex]\( x \)[/tex] starts from 0.
- End Point: The reasonable end point will depend on the context, indicating how far the ball can realistically travel and still be meaningful in the game. For practical purposes, in a soccer game, we need to find an upper limit to [tex]\( x \)[/tex] that makes sense.
### Step 3: Compare Given Options for Reasonable Domain
- Option 1: Reasonable domain: [tex]\( 0 \leq x \leq 25 \)[/tex]
- Option 2: Reasonable domain: [tex]\( 0 \leq x \leq 52 \)[/tex]
- Option 3: Reasonable domain: [tex]\( 0 \leq x \leq 25 \)[/tex]
- Option 4: Reasonable domain: [tex]\( 0 \leq x \leq 5 \)[/tex]
### Analysis of Options
We need to find an option where the mathematical domain is [tex]\( -\infty < x < \infty \)[/tex] and the reasonable domain makes practical sense.
Upon reviewing all given options:
- Mathematical Domain: We see that options 3 and 4 both state [tex]\( -\infty < x < \infty \)[/tex], which is correct for the mathematical domain.
- Reasonable for [tex]\( x \)[/tex]:
- Option 3 suggests a reasonable domain [tex]\( 0 \leq x \leq 25 \)[/tex], which is appropriate for a realistic distance in a soccer field scenario.
- Option 4 suggests [tex]\( 0 \leq x \leq 5 \)[/tex], which might be too short for practical purposes.
Given the context and practical applications, Option 3 is the most appropriate.
### Conclusion
Based on the above analysis, the mathematical domain and reasonable domain compare as follows:
- Mathematical Domain: [tex]\( -\infty < x < \infty \)[/tex]
- Reasonable Domain: [tex]\( 0 \leq x \leq 25 \)[/tex]
Thus, the correct answer is:
- Mathematical Domain: [tex]\( -\infty < x < \infty \)[/tex]
- Reasonable Domain: [tex]\( 0 \leq x \leq 25 \)[/tex]
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.
