91Ó°ÊÓ

Understanding the relationship between speed, distance, and time is crucial for solving any motion-related problem. In this kayaking scenario, the distance remains fixed at 4 km, but the total speed changes due to the current. The time taken for travel is derived from the formula: \ \( \text{time} = \frac{\text{distance}}{\text{speed}} \).

This formula means:

• Time (\( t \)) is directly proportional to distance (\( d \))
• Time (\( t \)) is inversely proportional to speed (\( v \))

In this problem:

• When traveling with the current: \( v = 4 + w \)
• When traveling against the current: \( v = 4 - w \)

Thus, time taken to cover 4 km:

• With the current: \( t = \frac{4}{4 + w} \)
• Against the current: \( t = \frac{4}{4 - w} \)

This understanding helps establish a clear link between speed, distance, and time in varying current conditions.

The domain and range of the function describe all possible values of the input (speed of the current) and the output (time taken, respectively). Here, we need to determine the domain of the function \( t(w) \).

For the function \( t(w) = \frac{4}{4 + w} \) or \( t(w) = \frac{4}{4 - w} \), \( w \) cannot be \( \pm 4 \) as these values make the denominator zero,
which leads to an undefined time:

• For \( w = 4 \), \( 4 + 4 = 0 \)
• For \( w = -4 \), \( 4 - 4 = 0 \)

Thus, we can define the domain as the set of all \( w \) values that lie between -4 and 4:
\[ -4 < w < 4 \]

The range, on the other hand, represents all possible time values. As the speed decreases towards 0 or increases towards -4 or 4, the time \( t \), gets very large. Thus, the range of the function is:\

• \( t > 0 \)

This means the time required will always be positive but it can get very large as the current speed approaches the kayak speed.

Understanding asymptotic behavior is important to analyze how the function behaves as the input values approach certain critical points. An asymptote is a line that the graph of a function approaches but never touches. In the context of this kayaking problem, the function \( t(w) \) displays vertical asymptotes at\

• \( w = 4 \)
• \( w = -4 \)

Analyzing the function \( \ t(w) = \frac{4}{4 + w} \) or \( t(w)= \frac{4}{4 - w} \), the critical points occur when \( w = 4 \) and \( w = -4 \).

As \( w \) approaches 4 or -4:

• The denominator of the function approaches zero
• The value of \( t(w) \) increases towards infinity

Graphically, this is shown as the curve getting closer and closer to the vertical line at \( w = 4 \) or \( w = -4 \), but never actually crossing it.
In practical terms for the kayaking scenario, if the current's speed reaches exactly 4 km/h either aiding or opposing the kayak's speed, they would theoretically take an infinite amount of time to travel 4 km, making normal kayaking impossible under these conditions.