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Which is true about the domains and ranges of the functions [tex]f(x) = \frac{1}{8} \sqrt{x}[/tex] and [tex]g(x) = 8 \sqrt{x}[/tex]?

A. They have the same domain and range.
B. They have different domains and different ranges.
C. They have the same range but different domains.
D. They have the same domain but different ranges.

Sagot :

GinnyAnswer
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.
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