Question

3.2.77 question 0/1 pt a. graph the function f(x) b. for x < 2, what is f(x)? c. for x > 2, what is f(x)? d. graph f on its domain e. is f differentiable at 2? explain f(x)={2x - 4 x≤2, 2x - 3 x>2} oa ob oc od

Explanation:

Step1: Recall derivative rules for piece - wise functions

For \(f(x)=

$$\begin{cases}2x - 4, &x\leq2\\2x-3, &x > 2\end{cases}$$

\), use the power rule \((x^n)^\prime=nx^{n - 1}\). For \(y = 2x-4\) where \(x\leq2\), \(y^\prime=f^\prime(x)=2\). For \(y = 2x - 3\) where \(x>2\), \(y^\prime=f^\prime(x)=2\).

Step2: Check differentiability at \(x = 2\)

Left - hand derivative: \(\lim_{x
ightarrow2^{-}}\frac{f(x)-f(2)}{x - 2}=\lim_{x
ightarrow2^{-}}\frac{(2x - 4)-(2\times2-4)}{x - 2}=\lim_{x
ightarrow2^{-}}\frac{2x - 4}{x - 2}=2\). Right - hand derivative: \(\lim_{x
ightarrow2^{+}}\frac{f(x)-f(2)}{x - 2}=\lim_{x
ightarrow2^{+}}\frac{(2x - 3)-(2\times2-4)}{x - 2}=\lim_{x
ightarrow2^{+}}\frac{2x - 3}{x - 2}\), but \(f(2)=2\times2 - 4=0\), \(\lim_{x
ightarrow2^{+}}\frac{2x - 3-0}{x - 2}\) is undefined in the context of the function's behavior at \(x = 2\) since the function changes its rule at \(x = 2\). The function is not differentiable at \(x = 2\) because the left - hand and right - hand limits of the difference quotient at \(x = 2\) do not match in a smooth way.

Step3: Answer parts of the question

a. To graph \(f(x)\), for \(y = 2x-4\) (\(x\leq2\)), it is a line with slope \(m = 2\) and \(y\) - intercept \((0,-4)\), and the point \((2,0)\) is included. For \(y = 2x-3\) (\(x>2\)), it is a line with slope \(m = 2\) and \(y\) - intercept \((0,-3)\), and the point \((2,1)\) is not included.
b. For \(x<2\), \(f^\prime(x)=2\).
c. For \(x>2\), \(f^\prime(x)=2\).
d. The domain of \(f^\prime(x)\) is \((-\infty,2)\cup(2,\infty)\).
e. \(f(x)\) is not differentiable at \(x = 2\) because the left - hand derivative and right - hand derivative at \(x = 2\) do not match smoothly (the function has a break in its definition at \(x = 2\)).

Answer:

a. Graph the line \(y = 2x-4\) for \(x\leq2\) (including the point \((2,0)\)) and \(y = 2x - 3\) for \(x>2\) (not including the point \((2,1)\)).
b. \(f^\prime(x)=2\) for \(x<2\).
c. \(f^\prime(x)=2\) for \(x>2\).
d. Domain of \(f^\prime(x)\) is \((-\infty,2)\cup(2,\infty)\).
e. \(f(x)\) is not differentiable at \(x = 2\) due to non - matching left and right - hand derivatives at \(x = 2\).