Question

If \( f(x) \) defined as given below, is continuous on \( R \), then the value of \( a + b \) is equal to: % Function Definition \[f(x) = \begin{cases} \sin x, & x \leq 0 \\ x^2 + a, & 0<x<1 \\ bx + 3, & 1 \leq x \leq 3 \\ -3, & x>3 \end{cases}\]

• A. 0
• B. 2
• C. -2
• D. 3

Hint:

For piecewise functions, check continuity at each boundary by equating the values of the function on both sides of the boundary.

Answer 1

The Correct Option is C

For the function to be continuous at the boundaries \( x = 0 \) and \( x = 1 \), the left-hand limit and the right-hand limit must be equal at these points. At \( x = 0 \): From the piecewise function, the left-hand side is: \[ \lim_{x \to 0^-} f(x) = \sin 0 = 0. \] The right-hand side is: \[ \lim_{x \to 0^+} f(x) = 0^2 + a = a. \] For continuity at \( x = 0 \): \[ a = 0. \] At \( x = 3 \): The left-hand limit is: \[ \lim_{x \to 3^-} f(x) = b(3) + 3 = 3b + 3. \] The right-hand limit is: \[ \lim_{x \to 3^+} f(x) = 3. \] For continuity at \( x = 3 \): \[ 3b + 3 = 3. \] Solving for \( b \): \[ 3b = 0 \Rightarrow b = -2. \] Final Calculation: \[ a + b = 0 + (-2) = -2. \] Final Answer: (c) -2.