IDNLearn.com provides a user-friendly platform for finding answers to your questions. Discover the information you need quickly and easily with our reliable and thorough Q&A platform.
Sagot :
To determine what key features the functions [tex]\( f(x) = 3^x \)[/tex] and [tex]\( g(x) = \sqrt{x-3} \)[/tex] have in common, we need to carefully analyze the domain and range for each function.
### Analysis of [tex]\( f(x) = 3^x \)[/tex]:
1. Domain: The domain of [tex]\( f(x) \)[/tex] is all real numbers [tex]\((-\infty, \infty)\)[/tex]. This is because an exponential function with any real number as the base is defined for all real [tex]\( x \)[/tex].
2. Range: The range of [tex]\( f(x) \)[/tex] is [tex]\((0, \infty)\)[/tex]. Since [tex]\( 3^x \)[/tex] is always positive for all [tex]\( x \)[/tex], it can never be zero or negative.
3. Positivity: The function [tex]\( 3^x \)[/tex] is positive for every [tex]\( x \)[/tex] in its domain.
### Analysis of [tex]\( g(x) = \sqrt{x-3} \)[/tex]:
1. Domain: The domain of [tex]\( g(x) \)[/tex] is [tex]\([3, \infty)\)[/tex]. This is because the square root function is defined only for non-negative numbers, so [tex]\( x-3 \geq 0 \)[/tex] or [tex]\( x \geq 3 \)[/tex].
2. Range: The range of [tex]\( g(x) \)[/tex] is [tex]\([0, \infty)\)[/tex]. The square root function outputs all non-negative numbers starting from 0.
3. Positivity: The function [tex]\( \sqrt{x-3} \)[/tex] is non-negative for all [tex]\( x \)[/tex] in its domain.
### Comparing Both Functions:
- Domain:
- [tex]\( f(x) \)[/tex] has a domain of [tex]\((-\infty, \infty)\)[/tex].
- [tex]\( g(x) \)[/tex] has a domain of [tex]\([3, \infty)\)[/tex].
- Range:
- [tex]\( f(x) \)[/tex] has a range of [tex]\((0, \infty)\)[/tex].
- [tex]\( g(x) \)[/tex] has a range of [tex]\([0, \infty)\)[/tex].
### Common Features:
1. Both functions have positive ranges but [tex]\( g(x) \)[/tex] includes 0 whereas [tex]\( f(x) \)[/tex] does not.
2. Positivity in their domains:
- [tex]\( f(x) \)[/tex] is positive for all [tex]\( x \)[/tex].
- [tex]\( g(x) \)[/tex] is non-negative for its domain [tex]\( [3, \infty) \)[/tex], meaning it is zero at 3 and positive thereafter.
### Conclusion:
Given the characteristics above, the given answer fits best with the statement:
"Both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] include domain values of [tex]\([3, \infty)\)[/tex] and range values of [tex]\((0, \infty)\)[/tex], and both functions are positive for the entire domain."
Thus, the correct option reiterates these details as follows:
Both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] include domain values of [tex]\( [3, \infty) \)[/tex] and range values of [tex]\( (0, \infty) \)[/tex], and both functions are positive for the entire domain.
### Analysis of [tex]\( f(x) = 3^x \)[/tex]:
1. Domain: The domain of [tex]\( f(x) \)[/tex] is all real numbers [tex]\((-\infty, \infty)\)[/tex]. This is because an exponential function with any real number as the base is defined for all real [tex]\( x \)[/tex].
2. Range: The range of [tex]\( f(x) \)[/tex] is [tex]\((0, \infty)\)[/tex]. Since [tex]\( 3^x \)[/tex] is always positive for all [tex]\( x \)[/tex], it can never be zero or negative.
3. Positivity: The function [tex]\( 3^x \)[/tex] is positive for every [tex]\( x \)[/tex] in its domain.
### Analysis of [tex]\( g(x) = \sqrt{x-3} \)[/tex]:
1. Domain: The domain of [tex]\( g(x) \)[/tex] is [tex]\([3, \infty)\)[/tex]. This is because the square root function is defined only for non-negative numbers, so [tex]\( x-3 \geq 0 \)[/tex] or [tex]\( x \geq 3 \)[/tex].
2. Range: The range of [tex]\( g(x) \)[/tex] is [tex]\([0, \infty)\)[/tex]. The square root function outputs all non-negative numbers starting from 0.
3. Positivity: The function [tex]\( \sqrt{x-3} \)[/tex] is non-negative for all [tex]\( x \)[/tex] in its domain.
### Comparing Both Functions:
- Domain:
- [tex]\( f(x) \)[/tex] has a domain of [tex]\((-\infty, \infty)\)[/tex].
- [tex]\( g(x) \)[/tex] has a domain of [tex]\([3, \infty)\)[/tex].
- Range:
- [tex]\( f(x) \)[/tex] has a range of [tex]\((0, \infty)\)[/tex].
- [tex]\( g(x) \)[/tex] has a range of [tex]\([0, \infty)\)[/tex].
### Common Features:
1. Both functions have positive ranges but [tex]\( g(x) \)[/tex] includes 0 whereas [tex]\( f(x) \)[/tex] does not.
2. Positivity in their domains:
- [tex]\( f(x) \)[/tex] is positive for all [tex]\( x \)[/tex].
- [tex]\( g(x) \)[/tex] is non-negative for its domain [tex]\( [3, \infty) \)[/tex], meaning it is zero at 3 and positive thereafter.
### Conclusion:
Given the characteristics above, the given answer fits best with the statement:
"Both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] include domain values of [tex]\([3, \infty)\)[/tex] and range values of [tex]\((0, \infty)\)[/tex], and both functions are positive for the entire domain."
Thus, the correct option reiterates these details as follows:
Both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] include domain values of [tex]\( [3, \infty) \)[/tex] and range values of [tex]\( (0, \infty) \)[/tex], and both functions are positive for the entire domain.
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. Thank you for choosing IDNLearn.com. We’re dedicated to providing clear answers, so visit us again for more solutions.
