Question

If $f(x) = 3x - b$, $x>1$ ; $f(x) = 11$, $x = 1$ ; $f(x) = -3x - 2b$, $x<1$ is continuous at $x = 1$, then the values of $a$ and $b$ are :

• A. $a = 3$, $b = 5$
• B. $a = 5$, $b = 3$
• C. $a = 8$, $b = 5$
• D. $a = -3$, $b = 5$

Hint:

For a piecewise function to be continuous, the function must have the same value from both sides at the point of interest.

Answer 1

The Correct Option is A

For the function to be continuous at $x = 1$, the limit of $f(x)$ as $x$ approaches 1 from both sides must be equal. \[ \lim_{x \to 1^+} f(x) = \lim_{x \to 1^-} f(x) = f(1) = 11 \] From the given piecewise function: For $x>1$, we have $f(x) = 3x - b$. At $x = 1$, \[ 3(1) - b = 11 \Rightarrow b = 5 \] Thus, the values of $a$ and $b$ are $a = 3$ and $b = 5$.