let f be the function given by f(x)=(x² - 3·(x + 0.5))/((x² - 3)(x + 0.5)). on which of the following open…

let f be the function given by f(x)=(x² - 3·(x + 0.5))/((x² - 3)(x + 0.5)). on which of the following open intervals is f continuous? a (-2,-1) b (-1,0) c (0,1) d (1,2)

let f be the function given by f(x)=(x² - 3·(x + 0.5))/((x² - 3)(x + 0.5)). on which of the following open intervals is f continuous? a (-2,-1) b (-1,0) c (0,1) d (1,2)

Answer

Explanation:

Step1: Find the domain - undefined points

A rational function $y = \frac{g(x)}{h(x)}$ is undefined when $h(x)=0$. Set $(x^{2}-3)(x + 0.5)=0$. We have two cases: $x^{2}-3=0$ or $x+0.5 = 0$. For $x^{2}-3=0$, then $x=\pm\sqrt{3}\approx\pm1.732$. For $x + 0.5=0$, then $x=- 0.5$.

Step2: Check each interval

  • For interval A $(-2,-1)$: The values $x =-\sqrt{3}\approx - 1.732$ is in this interval, so the function is not continuous on $(-2,-1)$.
  • For interval B $(-1,0)$: The value $x=-0.5$ is in this interval, so the function is not continuous on $(-1,0)$.
  • For interval C $(0,1)$: None of the values $x=\pm\sqrt{3},x = - 0.5$ are in the interval $(0,1)$. So the function is continuous on $(0,1)$.
  • For interval D $(1,2)$: The value $x=\sqrt{3}\approx1.732$ is in this interval, so the function is not continuous on $(1,2)$.

Answer:

C. $(0,1)$