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To determine which key features the functions [tex]\( f(x) = 3^x \)[/tex] and [tex]\( g(x) = \sqrt{x-3} \)[/tex] have in common, let's analyze their domains, ranges, and behaviors carefully.
### Function Analysis:
1. Function [tex]\( f(x) = 3^x \)[/tex]:
- Domain: Since [tex]\( f(x) = 3^x \)[/tex] is an exponential function, it can take any real number as input. Therefore, the domain is [tex]\( (-\infty, \infty) \)[/tex].
- Range: The function only produces positive values because [tex]\( 3^x \)[/tex] is strictly greater than 0 for all real [tex]\( x \)[/tex]. Hence, the range is [tex]\( (0, \infty) \)[/tex].
2. Function [tex]\( g(x) = \sqrt{x-3} \)[/tex]:
- Domain: The expression inside the square root, [tex]\( x-3 \)[/tex], must be non-negative. Thus, [tex]\( x-3 \geq 0 \rightarrow x \geq 3 \)[/tex]. Therefore, the domain is [tex]\( [3, \infty) \)[/tex].
- Range: The square root function outputs only non-negative values. Starting from [tex]\( x=3 \)[/tex], where [tex]\( g(3) = \sqrt{3-3} = 0 \)[/tex], and increasing without bound, the range is [tex]\( [0, \infty) \)[/tex].
### Key Features Comparison:
To determine what these functions have in common, let's compare their features:
- Domain:
- [tex]\( f(x) \)[/tex] has the domain [tex]\( (-\infty, \infty) \)[/tex].
- [tex]\( g(x) \)[/tex] has the domain [tex]\( [3, \infty) \)[/tex].
- Range:
- [tex]\( f(x) \)[/tex] has the range [tex]\( (0, \infty) \)[/tex].
- [tex]\( g(x) \)[/tex] has the range [tex]\( [0, \infty) \)[/tex].
From this analysis, it’s clear that they do not share the same domain entirely, but if we restrict the domain of [tex]\( f(x) \)[/tex] to [tex]\( [3, \infty) \)[/tex], we can observe the following:
- Restricted Domain: Both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have a domain starting at 3 ([tex]\( [3, \infty) \)[/tex]).
- Range: Both functions have ranges that consist of positive values: [tex]\( (0, \infty) \)[/tex] for [tex]\( f(x) \)[/tex] and [tex]\( [0, \infty) \)[/tex] for [tex]\( g(x) \)[/tex].
- Positivity: Both functions only produce positive values within their respective domains.
Given the information about their key features:
- Both functions include domain values of [tex]\( [3, \infty) \)[/tex].
- Both functions possess range values of [tex]\( [0, \infty) \)[/tex].
- Both functions are positive for their entire domain.
Therefore, the correct choice would be:
### Answer:
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.
### Function Analysis:
1. Function [tex]\( f(x) = 3^x \)[/tex]:
- Domain: Since [tex]\( f(x) = 3^x \)[/tex] is an exponential function, it can take any real number as input. Therefore, the domain is [tex]\( (-\infty, \infty) \)[/tex].
- Range: The function only produces positive values because [tex]\( 3^x \)[/tex] is strictly greater than 0 for all real [tex]\( x \)[/tex]. Hence, the range is [tex]\( (0, \infty) \)[/tex].
2. Function [tex]\( g(x) = \sqrt{x-3} \)[/tex]:
- Domain: The expression inside the square root, [tex]\( x-3 \)[/tex], must be non-negative. Thus, [tex]\( x-3 \geq 0 \rightarrow x \geq 3 \)[/tex]. Therefore, the domain is [tex]\( [3, \infty) \)[/tex].
- Range: The square root function outputs only non-negative values. Starting from [tex]\( x=3 \)[/tex], where [tex]\( g(3) = \sqrt{3-3} = 0 \)[/tex], and increasing without bound, the range is [tex]\( [0, \infty) \)[/tex].
### Key Features Comparison:
To determine what these functions have in common, let's compare their features:
- Domain:
- [tex]\( f(x) \)[/tex] has the domain [tex]\( (-\infty, \infty) \)[/tex].
- [tex]\( g(x) \)[/tex] has the domain [tex]\( [3, \infty) \)[/tex].
- Range:
- [tex]\( f(x) \)[/tex] has the range [tex]\( (0, \infty) \)[/tex].
- [tex]\( g(x) \)[/tex] has the range [tex]\( [0, \infty) \)[/tex].
From this analysis, it’s clear that they do not share the same domain entirely, but if we restrict the domain of [tex]\( f(x) \)[/tex] to [tex]\( [3, \infty) \)[/tex], we can observe the following:
- Restricted Domain: Both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have a domain starting at 3 ([tex]\( [3, \infty) \)[/tex]).
- Range: Both functions have ranges that consist of positive values: [tex]\( (0, \infty) \)[/tex] for [tex]\( f(x) \)[/tex] and [tex]\( [0, \infty) \)[/tex] for [tex]\( g(x) \)[/tex].
- Positivity: Both functions only produce positive values within their respective domains.
Given the information about their key features:
- Both functions include domain values of [tex]\( [3, \infty) \)[/tex].
- Both functions possess range values of [tex]\( [0, \infty) \)[/tex].
- Both functions are positive for their entire domain.
Therefore, the correct choice would be:
### Answer:
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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