Question

Let f be the function defined by f(x) = |x-2| for all x. which of the following statements is true?

a. f is continuous but not differentiable at x=2.
b. f is differentiable at x=2.
c. f is not continuous at x=2. f
d. x=2 is a vertical asymptote of the graph of f.

Answer 1

The correct statement is a. f(x) = |x-2| is continuous but not differentiable at x=2, because the graph of f(x) has a v-shaped bend at x=2 with different slopes on either side of this point.

Step-by-step explanation:

The student has asked which statement is true about the function f(x) = |x-2|. This function represents the absolute value of (x-2), which implies that the graph of f(x) will have a v-shaped bend at x=2.

The two important properties we need to consider here are continuity and differentiability.

Let us evaluate the function around the point x=2. The function can be written as f(x) = x-2 if x ≥ 2 and f(x) = -(x-2) if x < 2, breaking it into piecewise linear functions.

Regardless, the function takes the value 0 at x=2, creating a seamless transition from one piece to the other. This makes the function continuous at x=2.

However, examining the slope of the graph, we have a slope of 1 for x > 2 and -1 for x < 2. Thus, the slope changes abruptly at x=2, signaling a lack of differentiability at x=2.

There is no vertical asymptote at x=2 since the function is defined and continuous there.

Hence, the correct statement regarding the function is that f is continuous but not differentiable at x=2, which corresponds to option (a).