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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.
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