For what value of $α$, the function $f$ defined by $f(x) = \left\{\begin{matrix}a(x^2-2x+1),& if\, x ≤0\\2x+1,& if\, x > 0\end{matrix}\right.$ is continuous at $x = 0$?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → $α = 1$
Given: The function is
$f(x) = \begin{cases} a(x^2 - 2x + 1), & \text{if } x \leq 0 \\ 2x + 1, & \text{if } x > 0 \end{cases}$
To ensure continuity at $x = 0$, we must have:
$\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0)$
Left-hand limit (LHL):
$\lim_{x \to 0^-} f(x) = a(0^2 - 2 \cdot 0 + 1) = a(1) = a$
Right-hand limit (RHL):
$\lim_{x \to 0^+} f(x) = 2 \cdot 0 + 1 = 1$
Also, $f(0) = a(0^2 - 2 \cdot 0 + 1) = a$
For continuity: $a = 1$