Answer :
Let's consider the functions [tex]\( f(x) = \frac{1}{8} \sqrt{x} \)[/tex] and [tex]\( g(x) = 8 \sqrt{x} \)[/tex] and analyze their domains and ranges.
### Domains:
1. [tex]\( f(x) = \frac{1}{8} \sqrt{x} \)[/tex]
- The function [tex]\( f(x) \)[/tex] involves the square root of [tex]\( x \)[/tex]. Since the square root function [tex]\( \sqrt{x} \)[/tex] is only defined for [tex]\( x \geq 0 \)[/tex], the domain of [tex]\( f(x) \)[/tex] is [tex]\( x \geq 0 \)[/tex], or in interval notation, [tex]\([0, \infty)\)[/tex].
2. [tex]\( g(x) = 8 \sqrt{x} \)[/tex]
- Similarly, the function [tex]\( g(x) \)[/tex] also involves the square root of [tex]\( x \)[/tex]. Hence, the domain of [tex]\( g(x) \)[/tex] is also [tex]\( x \geq 0 \)[/tex], or in interval notation, [tex]\([0, \infty)\)[/tex].
### Ranges:
1. [tex]\( f(x) = \frac{1}{8} \sqrt{x} \)[/tex]
- For [tex]\( f(x) = \frac{1}{8} \sqrt{x} \)[/tex], as [tex]\( x \)[/tex] goes from [tex]\( 0 \)[/tex] to [tex]\( \infty \)[/tex], [tex]\( \sqrt{x} \)[/tex] also goes from [tex]\( 0 \)[/tex] to [tex]\( \infty \)[/tex]. Multiplying by [tex]\( \frac{1}{8} \)[/tex], the output [tex]\( f(x) \)[/tex] ranges from [tex]\( 0 \)[/tex] to [tex]\( \infty \)[/tex] multiplied by [tex]\( \frac{1}{8} \)[/tex], which results in:
[tex]\[ f(x) \in \left[0, \infty \right) \][/tex]
2. [tex]\( g(x) = 8 \sqrt{x} \)[/tex]
- For [tex]\( g(x) = 8 \sqrt{x} \)[/tex], as [tex]\( x \)[/tex] goes from [tex]\( 0 \)[/tex] to [tex]\( \infty \)[/tex], [tex]\( \sqrt{x} \)[/tex] also goes from [tex]\( 0 \)[/tex] to [tex]\( \infty \)[/tex]. Multiplying by [tex]\( 8 \)[/tex], the output [tex]\( g(x) \)[/tex] ranges from [tex]\( 0 \)[/tex] to [tex]\( \infty \)[/tex] multiplied by [tex]\( 8 \)[/tex], which results in:
[tex]\[ g(x) \in \left[0, \infty \right) \][/tex]
Although both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have the same domain [tex]\([0, \infty)\)[/tex], their ranges are different due to the different scaling factors involved ([tex]\(\frac{1}{8}\)[/tex] for [tex]\( f(x) \)[/tex] and [tex]\( 8 \)[/tex] for [tex]\( g(x)\)[/tex]). Thus, if we compare the ranges:
- [tex]\( f(x) = \frac{1}{8} \sqrt{x} \)[/tex] yields a smaller range scaled down by [tex]\( \frac{1}{8} \)[/tex].
- [tex]\( g(x) = 8 \sqrt{x} \)[/tex] yields a larger range scaled up by [tex]\( 8 \)[/tex].
Therefore, the correct statement is:
"They have the same domain but different ranges."
So, the answer is 4.
### Domains:
1. [tex]\( f(x) = \frac{1}{8} \sqrt{x} \)[/tex]
- The function [tex]\( f(x) \)[/tex] involves the square root of [tex]\( x \)[/tex]. Since the square root function [tex]\( \sqrt{x} \)[/tex] is only defined for [tex]\( x \geq 0 \)[/tex], the domain of [tex]\( f(x) \)[/tex] is [tex]\( x \geq 0 \)[/tex], or in interval notation, [tex]\([0, \infty)\)[/tex].
2. [tex]\( g(x) = 8 \sqrt{x} \)[/tex]
- Similarly, the function [tex]\( g(x) \)[/tex] also involves the square root of [tex]\( x \)[/tex]. Hence, the domain of [tex]\( g(x) \)[/tex] is also [tex]\( x \geq 0 \)[/tex], or in interval notation, [tex]\([0, \infty)\)[/tex].
### Ranges:
1. [tex]\( f(x) = \frac{1}{8} \sqrt{x} \)[/tex]
- For [tex]\( f(x) = \frac{1}{8} \sqrt{x} \)[/tex], as [tex]\( x \)[/tex] goes from [tex]\( 0 \)[/tex] to [tex]\( \infty \)[/tex], [tex]\( \sqrt{x} \)[/tex] also goes from [tex]\( 0 \)[/tex] to [tex]\( \infty \)[/tex]. Multiplying by [tex]\( \frac{1}{8} \)[/tex], the output [tex]\( f(x) \)[/tex] ranges from [tex]\( 0 \)[/tex] to [tex]\( \infty \)[/tex] multiplied by [tex]\( \frac{1}{8} \)[/tex], which results in:
[tex]\[ f(x) \in \left[0, \infty \right) \][/tex]
2. [tex]\( g(x) = 8 \sqrt{x} \)[/tex]
- For [tex]\( g(x) = 8 \sqrt{x} \)[/tex], as [tex]\( x \)[/tex] goes from [tex]\( 0 \)[/tex] to [tex]\( \infty \)[/tex], [tex]\( \sqrt{x} \)[/tex] also goes from [tex]\( 0 \)[/tex] to [tex]\( \infty \)[/tex]. Multiplying by [tex]\( 8 \)[/tex], the output [tex]\( g(x) \)[/tex] ranges from [tex]\( 0 \)[/tex] to [tex]\( \infty \)[/tex] multiplied by [tex]\( 8 \)[/tex], which results in:
[tex]\[ g(x) \in \left[0, \infty \right) \][/tex]
Although both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have the same domain [tex]\([0, \infty)\)[/tex], their ranges are different due to the different scaling factors involved ([tex]\(\frac{1}{8}\)[/tex] for [tex]\( f(x) \)[/tex] and [tex]\( 8 \)[/tex] for [tex]\( g(x)\)[/tex]). Thus, if we compare the ranges:
- [tex]\( f(x) = \frac{1}{8} \sqrt{x} \)[/tex] yields a smaller range scaled down by [tex]\( \frac{1}{8} \)[/tex].
- [tex]\( g(x) = 8 \sqrt{x} \)[/tex] yields a larger range scaled up by [tex]\( 8 \)[/tex].
Therefore, the correct statement is:
"They have the same domain but different ranges."
So, the answer is 4.
