To determine the behavior of the function [tex]\( h(x) = 3 \sqrt{x+1} - 2 \)[/tex], we need to analyze in which intervals the function is increasing or decreasing.
1. Understanding the Function [tex]\( h(x) \)[/tex]:
[tex]\[
h(x) = 3 \sqrt{x+1} - 2
\][/tex]
The function involves a square root, which means it is defined for [tex]\( x \geq -1 \)[/tex].
2. Finding the Derivative [tex]\( h'(x) \)[/tex]:
To understand how the function behaves, we need to find its derivative [tex]\( h'(x) \)[/tex]:
[tex]\[
h(x) = 3 (x+1)^{1/2} - 2
\][/tex]
Differentiating this with respect to [tex]\( x \)[/tex]:
[tex]\[
h'(x) = 3 \cdot \frac{1}{2} (x+1)^{-1/2} \cdot 1 = \frac{3}{2} (x+1)^{-1/2} = \frac{3}{2\sqrt{x+1}}
\][/tex]
3. Analyzing the Derivative [tex]\( h'(x) \)[/tex]:
The sign of the derivative will tell us whether the function is increasing or decreasing.
[tex]\[
h'(x) = \frac{3}{2\sqrt{x+1}}
\][/tex]
For [tex]\( x \geq -1 \)[/tex]:
- The term [tex]\( \sqrt{x+1} \)[/tex] is always positive since [tex]\( x+1 \geq 0 \)[/tex].
- The fraction [tex]\( \frac{3}{2\sqrt{x+1}} \)[/tex] is always positive for [tex]\( x \geq -1 \)[/tex].
Hence, the derivative [tex]\( h'(x) \)[/tex] is positive for all [tex]\( x \geq -1 \)[/tex], indicating that the function is increasing in this interval.
4. Conclusion:
Since the derivative [tex]\( h'(x) \)[/tex] is positive for all [tex]\( x \geq -1 \)[/tex], the function [tex]\( h(x) \)[/tex] is increasing on the interval [tex]\( (-1, \infty) \)[/tex].
Therefore, the correct statement is:
- The function is increasing on the interval [tex]\( (-1, \infty) \)[/tex].
So, the answer is:
[tex]\[
\boxed{\text{The function is increasing on the interval } (-1, \infty).}
\][/tex]
