Question

The relationship between \( a \) and \( b \) for the continuity of the function \( f(x) = \begin{cases} ax + 1, & x \leq 3 \\ bx + 3, & x \gt 3 \end{cases} \) at \( x = 3 \) is __________.

• A. \( 3b = 3a + 2 \)
• B. \( 3a = b + 2 \)
• C. \( 3a = 3b + 2 \)
• D. \( a = 3b - 2 \)

Hint:

For piecewise functions, continuity at a point means left limit equals right limitAlways equate both expressions at that point.

Answer 1

The Correct Option is C

Step 1: Condition for continuity.
For continuity at \( x = 3 \):
\[ \text{LHL} = \text{RHL} \]

Step 2: Compute LHL.
\[ f(3) = 3a + 1 \]

Step 3: Compute RHL.
\[ \lim_{x \to 3^+} f(x) = 3b + 3 \]

Step 4: Apply continuity condition.
\[ 3a + 1 = 3b + 3 \]

Step 5: Simplify equation.
\[ 3a = 3b + 2 \]

Step 6: Match with options.
Correct option is (C).

Step 7: Final conclusion.
\[ \boxed{3a = 3b + 2} \]