To determine the range of the linear function [tex]\( h(x) = -2x + 35 \)[/tex] which models the height of the river, let's analyze the function step-by-step.
1. Identify the Function Type:
The given function is a linear function of the form [tex]\( h(x) = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- Here, [tex]\( m = -2 \)[/tex] and [tex]\( b = 35 \)[/tex].
2. Understand the Behavior of the Function:
- The slope [tex]\( m = -2 \)[/tex] is negative, indicating that the function is decreasing. As the number of days [tex]\( x \)[/tex] increases, the value of [tex]\( h(x) \)[/tex] decreases.
3. Analyze Critical Points:
- When [tex]\( x = 0 \)[/tex], the height of the river [tex]\( h(0) = -2(0) + 35 = 35 \)[/tex] feet.
4. Determine Limits as [tex]\( x \)[/tex] Changes:
- As [tex]\( x \)[/tex] increases (approaches positive infinity), the term [tex]\(-2x\)[/tex] becomes very large negative, making [tex]\( h(x) \)[/tex] approach negative infinity.
- As [tex]\( x \)[/tex] decreases (approaches negative infinity), the term [tex]\(-2x\)[/tex] becomes very large positive, but since we are considering real, practical values of [tex]\( x \)[/tex] representing days, the primary consideration is non-negative days (where [tex]\( x \geq 0 \)[/tex]).
5. Find the Range:
- Given real-world non-negative [tex]\( x \)[/tex] values (days since the observation started), the function [tex]\( h(x) \)[/tex] can take any value starting from the initial height of 35 feet down to smaller values as days progress.
- It implies [tex]\( h(x) \)[/tex] can range from the maximum height of 35 feet down to negative infinity as [tex]\( x \)[/tex] increases (i.e., more days pass).
Therefore, the range of the function [tex]\( h(x) \)[/tex] is [tex]\( (-\infty, 35] \)[/tex].
The correct multiple-choice answer is:
- [tex]\(\boxed{(-\infty, 35]}\)[/tex]
