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

Answer 1

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

1. Differentiate the function: The derivative of the function \( y = 3x - 1 - \sqrt{3} \) is simply the derivative of the linear component since the constants have a derivative of zero.

   \[ \frac{dy}{dx} = \frac{d}{dx}(3x) - \frac{d}{dx}(1) - \frac{d}{dx}(\sqrt{3}) = 3 - 0 - 0 = 3 \]

2. Analyze the derivative: The derivative \( \frac{dy}{dx} = 3 \) is positive. This means that for any value of \( x \), the function is increasing because the slope of the tangent line to the curve is positive.

Conclusion: The graph of the function shows that the function is increasing.

Final Answer: Option 1: increasing