To determine the behavior of the function [tex]\( h(x) = -5 \sqrt{x} \)[/tex], let’s analyze it step by step.
1. Domain of the Function:
The function [tex]\( \sqrt{x} \)[/tex] is only defined for [tex]\( x \geq 0 \)[/tex] because the square root of a negative number is not a real number. Therefore, the domain of [tex]\( h(x) = -5 \sqrt{x} \)[/tex] is [tex]\( x \geq 0 \)[/tex].
2. Behavior of the Square Root Function:
The fundamental parent function [tex]\( \sqrt{x} \)[/tex] is increasing on its entire domain [tex]\( x \geq 0 \)[/tex]. As [tex]\( x \)[/tex] increases, [tex]\( \sqrt{x} \)[/tex] also increases.
3. Effect of the Multiplication by -5:
Multiplying the square root function by a negative constant, in this case, -5, reverses its direction. Thus, instead of increasing, the function [tex]\( h(x) = -5 \sqrt{x} \)[/tex] will be decreasing on its domain. The negative sign indicates that as [tex]\( x \)[/tex] increases, [tex]\( -5 \sqrt{x} \)[/tex] decreases.
4. Interval Analysis:
Since [tex]\( h(x) = -5 \sqrt{x} \)[/tex] is defined for [tex]\( x \geq 0 \)[/tex], we need to consider the intervals within this domain. Analyzing the behavior of the function within the interval [tex]\( (0, \infty) \)[/tex]:
- As [tex]\( x \)[/tex] increases from 0 to [tex]\( \infty \)[/tex], [tex]\( \sqrt{x} \)[/tex] increases, but multiplying by -5 makes the entire expression [tex]\( -5 \sqrt{x} \)[/tex] decrease.
Therefore, we conclude that the function [tex]\( h(x) = -5 \sqrt{x} \)[/tex] is decreasing on the interval [tex]\( (0, \infty) \)[/tex].
Among the given statements, the correct one is:
- The function is decreasing on the interval [tex]\( (0, \infty) \)[/tex].
