Sure, let's delve into the key features of the functions [tex]\( f(x) = 3^x \)[/tex] and [tex]\( g(x) = \sqrt{x-3} \)[/tex].
1. Analyzing [tex]\( f(x) = 3^x \)[/tex]:
- Domain: The domain of [tex]\( f(x) = 3^x \)[/tex] is all real numbers. This is because you can raise 3 to any power, whether the exponent is positive, negative, or zero. Therefore, the domain is [tex]\( (-\infty, \infty) \)[/tex].
- Range: The range of [tex]\( f(x) = 3^x \)[/tex] is all positive real numbers. As [tex]\( x \)[/tex] becomes very large, [tex]\( 3^x \)[/tex] grows towards infinity. As [tex]\( x \)[/tex] becomes very small (i.e., very negative), [tex]\( 3^x \)[/tex] approaches zero but never actually reaches zero. Hence, the range is [tex]\( (0, \infty) \)[/tex].
2. Analyzing [tex]\( g(x) = \sqrt{x-3} \)[/tex]:
- Domain: The domain of [tex]\( g(x) = \sqrt{x-3} \)[/tex] consists of all [tex]\( x \)[/tex] values that make the expression inside the square root non-negative. This occurs when [tex]\( x - 3 \geq 0 \)[/tex], or [tex]\( x \geq 3 \)[/tex]. Therefore, the domain is [tex]\( [3, \infty) \)[/tex].
- Range: The range of [tex]\( g(x) = \sqrt{x-3} \)[/tex] is all non-negative real numbers. The smallest value [tex]\( g(x) \)[/tex] can take is 0, which happens when [tex]\( x = 3 \)[/tex]. As [tex]\( x \)[/tex] increases, [tex]\( \sqrt{x-3} \)[/tex] also increases without bound. Therefore, the range is [tex]\( [0, \infty) \)[/tex].
3. Comparing the key features:
- Same Range: Both functions share a similar range. For [tex]\( f(x) = 3^x \)[/tex], the range is [tex]\( (0, \infty) \)[/tex], and for [tex]\( g(x) = \sqrt{x-3} \)[/tex], the range is [tex]\( [0, \infty) \)[/tex]. Despite the slight difference at 0 (where [tex]\( f(x) \)[/tex] does not include 0 but [tex]\( g(x) \)[/tex] does include 0), both functions map to positive values as [tex]\( x \)[/tex] grows.
- Different Domains: The domain of [tex]\( f(x) = 3^x \)[/tex] is all real numbers [tex]\( (-\infty, \infty) \)[/tex], while the domain of [tex]\( g(x) = \sqrt{x-3} \)[/tex] is [tex]\( [3, \infty) \)[/tex].
Conclusion:
The key feature the functions [tex]\( f(x) = 3^x \)[/tex] and [tex]\( g(x) = \sqrt{x-3} \)[/tex] have in common is their range, which is [tex]\( (0, \infty) \)[/tex] (taking into account that [tex]\( g(x) \)[/tex] includes 0, its range is technically [tex]\( [0, \infty) \)[/tex]). However, they have different domains.
