Question

The number of points of discontinuity of \[ f(x) = \begin{cases} |x| + 3, & \text{if } x \leq -3, \\ -2x, & \text{if } -3 < x < 3, \\ 6x + 2, & \text{if } x \geq 3 \end{cases} \] is:

• A. \( 0 \)
• B. \( 1 \)
• C. \( 2 \)
• D. infinite

Hint:

To check discontinuity at a point, evaluate the left-hand limit, right-hand limit, and the functional value. If any of these do not match, the function is discontinuous at that point.

Answer 1

The Correct Option is B

Step 1: Identify the points of possible discontinuity. 
The given piecewise function has transitions at \( x = -3 \) and \( x = 3 \). These are the points where the function could be discontinuous. 
Step 2: Check continuity at \( x = -3 \). 
At \( x = -3 \): \[ \text{Left-hand limit (LHL)} = |x| + 3 = |-3| + 3 = 3 + 3 = 6, \] \[ \text{Right-hand limit (RHL)} = -2x = -2(-3) = 6. \] The functional value: \[ f(-3) = |x| + 3 = |-3| + 3 = 6. \] Since \( \text{LHL} = \text{RHL} = f(-3) \), \( f(x) \) is continuous at \( x = -3 \). 
Step 3: Check continuity at \( x = 3 \). 
At \( x = 3 \): \[ \text{Left-hand limit (LHL)} = -2x = -2(3) = -6, \] \[ \text{Right-hand limit (RHL)} = 6x + 2 = 6(3) + 2 = 18 + 2 = 20. \] Since \( \text{LHL} \neq \text{RHL} \), \( f(x) \) is discontinuous at \( x = 3 \). 
Step 4: Conclusion. 
There is only one point of discontinuity, which is at \( x = 3 \). The number of points of discontinuity is: \[ \boxed{1}. \]