inspect the graph of the function to determine whether it is increasing or decreasing on the given…

inspect the graph of the function to determine whether it is increasing or decreasing on the given interval.\n\n$f(x)=3 - \\sqrt{x}$ on $(0,\\infty)$\n\nchoose the correct answer below.\n\noa. the function is increasing on $(0,\\infty)$ because its graph is a curve that lies mostly above the x - axis.\n\nob. the function is decreasing on $(0,\\infty)$ because its graph is a curve that falls from left to right.\n\noc. the function is increasing on $(0,\\infty)$ because its graph is a curve that rises from left to right.\n\nod. the function is decreasing on $(0,\\infty)$ because its graph is a curve that lies mostly below the x - axis.

inspect the graph of the function to determine whether it is increasing or decreasing on the given interval.\n\n$f(x)=3 - \\sqrt{x}$ on $(0,\\infty)$\n\nchoose the correct answer below.\n\noa. the function is increasing on $(0,\\infty)$ because its graph is a curve that lies mostly above the x - axis.\n\nob. the function is decreasing on $(0,\\infty)$ because its graph is a curve that falls from left to right.\n\noc. the function is increasing on $(0,\\infty)$ because its graph is a curve that rises from left to right.\n\nod. the function is decreasing on $(0,\\infty)$ because its graph is a curve that lies mostly below the x - axis.

Answer

Explanation:

Step1: Find the derivative of the function

Given ( f(x)=\sqrt{x}-3), rewrite it as ( f(x)=x^{\frac{1}{2}} - 3). Using the power rule ( (x^n)^\prime=nx^{n - 1}), the derivative ( f^\prime(x)=\frac{1}{2}x^{-\frac{1}{2}}=\frac{1}{2\sqrt{x}}).

Step2: Analyze the sign of the derivative on the interval ((0,\infty))

For (x\in(0,\infty)), (\sqrt{x}>0). Then (f^\prime(x)=\frac{1}{2\sqrt{x}}>0) on the interval ((0,\infty)). When (f^\prime(x)>0) on an interval, the function (y = f(x)) is increasing on that interval. Since (y = f(x)) is increasing on ((0,\infty)), as (x) increases (moves from left to right) on ((0,\infty)), (y=f(x)) also increases.

Answer:

C. The function is increasing on ((0,\infty)) because its graph is a curve that rises from left to right.