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: \[ \boxed{1}. \]
Draw a rough sketch for the curve $y = 2 + |x + 1|$. Using integration, find the area of the region bounded by the curve $y = 2 + |x + 1|$, $x = -4$, $x = 3$, and $y = 0$.
A stationary tank is cylindrical in shape with two hemispherical ends and is horizontal, as shown in the figure. \(R\) is the radius of the cylinder as well as of the hemispherical ends. The tank is half filled with an oil of density \(\rho\) and the rest of the space in the tank is occupied by air. The air pressure, inside the tank as well as outside it, is atmospheric. The acceleration due to gravity (\(g\)) acts vertically downward. The net horizontal force applied by the oil on the right hemispherical end (shown by the bold outline in the figure) is: