Does the graph of the function y=3x−1−−−−−√3 show that the function is increasing or decreasing?

Question

Option 1: increasing

Option 2: decreasing

(1 point)
The correct answer is option
.

Answer 1

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

First, we need to differentiate the function:

1. Find the derivative: The function can be rewritten in a simpler form for differentiation as \( y = (3x - 1)^{1/2} \).

   Using the chain rule, the derivative \( \frac{dy}{dx} \) is: \[ \frac{dy}{dx} = \frac{1}{2}(3x - 1)^{-1/2} \cdot 3 = \frac{3}{2\sqrt{3x - 1}} \]

2. Analyze the derivative: The expression \( \frac{3}{2\sqrt{3x - 1}} \) is positive as long as \( 3x - 1 > 0 \) (which means \( x > \frac{1}{3} \)).

Since the derivative is positive for all \( x > \frac{1}{3} \), the function is increasing in that interval.

Thus, the correct answer is:

Option 1: increasing