Does the graph of the function y=3(x−1)−−−−−−√3

Question

show that the function is increasing or decreasing?(1 point)
Responses

increasing
increasing

decreasing
decreasing

Neither increasing nor decreasing
Neither increasing nor decreasing

Both increasing and decreasing
Both increasing and decreasing

Answer 1

To determine whether the function \( y = 3\sqrt{x-1} \) is increasing or decreasing, we can analyze the behavior of the function as \( x \) changes.

1. Domain: The function is defined for \( x \geq 1 \) (because of the square root).

2. Derivative: To find out if the function is increasing or decreasing, we can look at the derivative of the function:

   \[ y = 3\sqrt{x-1} \]

   Using the chain rule to differentiate:

   \[ y' = 3 \cdot \frac{1}{2\sqrt{x-1}} \cdot (1) = \frac{3}{2\sqrt{x-1}} \]

3. Sign of the Derivative: The derivative \( y' = \frac{3}{2\sqrt{x-1}} \) is positive for all \( x > 1 \). Since the derivative is positive in the domain \( x \geq 1 \), this means that the function is increasing in this interval.

Conclusion: The graph of the function \( y = 3\sqrt{x-1} \) shows that the function is increasing.

So the correct response is:

increasing.