suppose that the function f is defined, for all real numbers, as follows. f(x)={x - 4 if x<2 - 2x + 2 if x≥2…

suppose that the function f is defined, for all real numbers, as follows. f(x)={x - 4 if x<2 - 2x + 2 if x≥2 graph the function f. then determine whether or not the function is continuous. is the function continuous? yes no

suppose that the function f is defined, for all real numbers, as follows. f(x)={x - 4 if x<2 - 2x + 2 if x≥2 graph the function f. then determine whether or not the function is continuous. is the function continuous? yes no

Answer

Explanation:

Step1: Analyze $f(x)=x - 4$ for $x<2$

When $x = 2$, $y=x - 4=2 - 4=-2$. But for $x<2$, we have an open - circle at the point $(2,-2)$ on the line $y=x - 4$. The slope of the line $y=x - 4$ is $m = 1$ and the $y$ - intercept is $b=-4$.

Step2: Analyze $f(x)=-2x + 2$ for $x\geq2$

When $x = 2$, $y=-2x + 2=-2\times2+2=-2$. We have a closed - circle at the point $(2,-2)$ on the line $y=-2x + 2$. The slope of the line $y=-2x + 2$ is $m=-2$ and the $y$ - intercept is $b = 2$.

Step3: Check continuity

We find the left - hand limit as $x\to2^{-}$: $\lim_{x\to2^{-}}f(x)=\lim_{x\to2^{-}}(x - 4)=2-4=-2$. The right - hand limit as $x\to2^{+}$: $\lim_{x\to2^{+}}f(x)=\lim_{x\to2^{+}}(-2x + 2)=-2\times2+2=-2$. And $f(2)=-2\times2 + 2=-2$. Since $\lim_{x\to2^{-}}f(x)=\lim_{x\to2^{+}}f(x)=f(2)$, the function is continuous.

Answer:

Yes