Question

If $f(x) = \begin{cases} ax + b, & x \leq 3 \\ 7, & 5<x \end{cases}$ is continuous in $\mathbb{R}$, then the values of $a$ and $b$ are:

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

Hint:

For piecewise functions to be continuous, ensure the limits from both sides match at the point of transition.

Answer 1

The Correct Option is C

For the function to be continuous at $x = 3$ and $x = 5$, the left-hand and right-hand limits must be equal. At $x = 3$, we have: \[ a(3) + b = 7 \quad \text{(since it must match the second part of the function)} \] Thus, $3a + b = 7$. Solving for the second condition at $x = 5$: \[ a(5) + b = 7 \quad \text{(since it must match the second part of the function again)} \] Solving these equations will yield $a = -3$ and $b = -8$.