Question

If f(x) = { ax + 3, for x ≤ 2
a(x - 1), for x > 2 }, then the values of a for which f is continuous for all x are:

• A. 1 and -2
• B. 1 and 2
• C. -1 and 2
• D. -1 and -2

Hint:

For continuity of piecewise functions, ensure that the limits from both sides at the point of interest are equal.

Answer 1

The Correct Option is C

For \( f(x) \) to be continuous for all \( x \), the left-hand limit at \( x = 2 \) must equal the right-hand limit at \( x = 2 \). 1. For \( x \leq 2 \), \( f(x) = ax + 3 \). The left-hand limit as \( x \to 2^- \) is: \[ \lim_{x \to 2^-} f(x) = 2a + 3 \] 2. For \( x > 2 \), \( f(x) = a(x - 1) \). The right-hand limit as \( x \to 2^+ \) is: \[ \lim_{x \to 2^+} f(x) = a(2 - 1) = a \] 3. For continuity at \( x = 2 \), these two limits must be equal: \[ 2a + 3 = a \quad \Rightarrow \quad a = -3 \] Thus, \( a = -1 \) and \( a = 2 \) are the values of \( a \) that make \( f(x) \) continuous.