Select the correct answer.
Over which interval of the domain is function [tex]$h$[/tex] decreasing?
[tex]$h(x)=\left\{\begin{array}{ll} 2^x, & x\ \textless \ 1 \\ \sqrt{x+3}, & x \geq 1 \end{array}\right.$[/tex]
A. [tex]$(-\infty, 1)$[/tex]
B. The function is increasing only.
C. [tex]$(1, \infty)$[/tex]
D. [tex]$(-\infty, \infty)$[/tex]
Answer (1)
Analyze the function h ( x ) = 2 x for x < 1 , which is an increasing exponential function.
Analyze the function h ( x ) = x + 3 for x ≥ 1 , which is an increasing square root function.
Conclude that since both parts of the piecewise function are increasing, the function is increasing throughout its domain.
The function is increasing only, so the answer is B .
Explanation
Analyzing the piecewise function We are given a piecewise function h ( x ) and asked to determine the interval over which it is decreasing. The function is defined as:
The function definition h ( x ) = { 2 x , x + 3 , x < 1 x ≥ 1
Analyzing the first interval First, let's analyze the interval x < 1 , where h ( x ) = 2 x . This is an exponential function with a base of 2, which is greater than 1. Exponential functions with a base greater than 1 are always increasing. Therefore, h ( x ) = 2 x is increasing on the interval ( − ∞ , 1 ) .
Analyzing the second interval Next, let's analyze the interval x ≥ 1 , where h ( x ) = x + 3 . This is a square root function. As x increases, x + 3 also increases, and therefore x + 3 increases. Thus, h ( x ) = x + 3 is increasing on the interval [ 1 , ∞ ) .
Conclusion Since h ( x ) is increasing on both intervals ( − ∞ , 1 ) and [ 1 , ∞ ) , the function is increasing throughout its domain. Therefore, there is no interval over which h ( x ) is decreasing.
Final Answer The function h ( x ) is increasing only.
Examples
Understanding increasing and decreasing functions is crucial in many real-world applications. For example, in economics, you might analyze the growth of an investment over time. If the function representing the investment's value is increasing, it means your investment is growing. Conversely, if it were decreasing, it would indicate a loss. Similarly, in physics, you could analyze the speed of an object. An increasing function would mean the object is accelerating, while a decreasing function would mean it's decelerating. These concepts help in making informed decisions and predictions in various fields.
