Question

If a continuous function \( f \) is defined as \[ f(x) = \left\{ \begin{array}{ll} ax + 1, & x < 2 \\ x^2 + 7, & x \geq 2 \end{array} \right. \] then the value of \( a \) is:

• A. 7
• B. 6
• C. 5
• D. 3
• E. 2

Hint:

For continuous functions, ensure that the left-hand and right-hand limits are equal at the point of interest.

Answer 1

The Correct Option is C

Step 1: For \( f(x) \) to be continuous at \( x = 2 \), the left-hand limit and the right-hand limit must be equal. 
Hence, we equate the two functions at \( x = 2 \): \[ \lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x). \] Step 2: The left-hand limit for \( x<2 \) is \( f(x) = ax + 1 \), so: \[ \lim_{x \to 2^-} f(x) = 2a + 1. \] The right-hand limit for \( x \geq 2 \) is \( f(x) = x^2 + 7 \), so: \[ \lim_{x \to 2^+} f(x) = 2^2 + 7 = 4 + 7 = 11. \] Step 3: Equating both limits for continuity: \[ 2a + 1 = 11. \] Step 4: Solving for \( a \): \[ 2a = 10 \quad \Rightarrow \quad a = 5. \]