question 33 (1 point) which of the following statements is not true of f(x)=sqrt(x^2 - 25)? a f(x) is…

question 33 (1 point) which of the following statements is not true of f(x)=sqrt(x^2 - 25)? a f(x) is continuous at x=10 b f(x) is continuous on the interval 5, infinity) c f(x) is continuous on the interval -5,5 d f(x) is continuous on the interval (-infinity,-5 e none of the above

Answer

Explanation:

Step1: Determine domain of f(x)

For $f(x)=\sqrt{x^{2}-25}$, we need $x^{2}-25\geq0$. Solving $(x - 5)(x + 5)\geq0$, we get $x\leq - 5$ or $x\geq5$.

Step2: Analyze continuity

At $x = 10$, $f(10)=\sqrt{10^{2}-25}=\sqrt{75}$, and $\lim_{x\rightarrow10}f(x)=f(10)$, so $f(x)$ is continuous at $x = 10$.
On the interval $[5,\infty)$, for any $x_0\in[5,\infty)$, $\lim_{x\rightarrow x_0}f(x)=f(x_0)$, so $f(x)$ is continuous on $[5,\infty)$.
On the interval $[-5,5]$, when $x\in(- 5,5)$, $x^{2}-25<0$ and $f(x)$ is not a real - valued function. So $f(x)$ is not continuous on $[-5,5]$.
On the interval $(-\infty,-5]$, for any $x_0\in(-\infty,-5]$, $\lim_{x\rightarrow x_0}f(x)=f(x_0)$, so $f(x)$ is continuous on $(-\infty,-5]$.

Answer:

C. f(x) is continuous on the interval [-5,5]