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 determine if a function is continuous at a point, check if the left-hand limit, right-hand limit, and the function value all match. If they do not, the function is discontinuous at that point.

Answer 1

The Correct Option is B

Step 1: Analyze the points where discontinuity might occur. 
The given piecewise function transitions at \( x = -3 \) and \( x = 3 \). These are the potential points where discontinuities could arise. 
Step 2: Verify continuity at \( x = -3 \). 
At \( x = -3 \), calculate the left-hand and right-hand limits: \[ \text{Left-hand limit (LHL)} = |x| + 3 = |-3| + 3 = 3 + 3 = 6, \] \[ \text{Right-hand limit (RHL)} = -2x = -2(-3) = 6. \] The function value at \( x = -3 \) is: \[ f(-3) = |x| + 3 = |-3| + 3 = 6. \] Since the left-hand limit, right-hand limit, and the function's value at \( x = -3 \) are all equal, the function is continuous at \( x = -3 \). 
Step 3: Verify continuity at \( x = 3 \). 
At \( x = 3 \), calculate the left-hand and right-hand limits: \[ \text{Left-hand limit (LHL)} = -2x = -2(3) = -6, \] \[ \text{Right-hand limit (RHL)} = 6x + 2 = 6(3) + 2 = 18 + 2 = 20. \] Since the left-hand limit and right-hand limit are not equal, the function is discontinuous at \( x = 3 \). 
Step 4: Final Conclusion. 
There is exactly one point of discontinuity, which occurs at \( x = 3 \). Therefore, the total number of points of discontinuity is: \[ \boxed{1}. \]