Question

If the function \(f(x)\) defined by \(f(x) = \{\begin{array}{cc}ax+1 & \text{if }x\leq 3 \\ bx+3 & \text{if }x > 3\end{array}\) is continuous at \(x = 3\), then \((a-b) =\) ..........

• A. \(\frac{2}{3}\)
• B. \(\frac{3}{2}\)
• C. \(2\)
• D. \(3\)

Hint:

Continuity at x = 3 means the left and right values are equal.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
A function is continuous at \(x=3\) if the left-hand limit, the right-hand limit and the value at 3 are all equal.

Step 2: Evaluate the pieces
Left side and value: \(f(3)=3a+1\).
Right-hand limit: \(\displaystyle\lim_{x\to3^+}(bx+3)=3b+3\).

Step 3: Equate
\[ 3a+1=3b+3 \]
\[ 3a-3b=2\Rightarrow a-b=\frac23 \]

Step 4: Check the options
The result is \(\frac23\), option (A). Options (C) and (D) come from forgetting to divide by 3.

Final Answer:
Equating the two pieces at x = 3 gives a - b = 2/3, option (A). \[ \boxed{\frac{2}{3}} \]