Answer :
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]
