suppose that the function f is defined, for all real numbers, as follows. f(x)={x - 3 if x≤2 - 5x + 4 if x>2…

suppose that the function f is defined, for all real numbers, as follows. f(x)={x - 3 if x≤2 - 5x + 4 if x>2 graph the function f. then determine whether or not the function is continuous. is the function continuous? oyes ono

Answer

Explanation:

Step1: Find left - hand limit

As $x\to2^{-}$, $f(x)=x - 3$. So $\lim_{x\to2^{-}}f(x)=\lim_{x\to2^{-}}(x - 3)=2-3=-1$.

Step2: Find right - hand limit

As $x\to2^{+}$, $f(x)=-5x + 4$. So $\lim_{x\to2^{+}}f(x)=\lim_{x\to2^{+}}(-5x + 4)=-5\times2+4=-6$.

Step3: Find function value at $x = 2$

When $x = 2$, $f(x)=x - 3$, so $f(2)=2-3=-1$.

Step4: Check continuity

Since $\lim_{x\to2^{-}}f(x)=-1$, $\lim_{x\to2^{+}}f(x)=-6$ and $\lim_{x\to2^{-}}f(x)\neq\lim_{x\to2^{+}}f(x)$, the function is not continuous.