IGNOU MTE 3 2018 - Determine the value of the constant a if the function f (x) defined be

Question

Determine the value of the constant a if the function f (x) defined below is continuous at x = 2.

Image ignouacademy-com--p-constant-90321

02 Feb 2018

Answer :

Word Count : 359

To determine the value of the constant \( a \) such that the function \( f(x) \) is continuous at \( x = 2 \), we need to ensure that the left-hand limit, right-hand limit, and the value of the function at \( x = 2 \) are all equal. Here's the step-by-step solution: ### Given Function The function \( f(x) \) is defined as: \[ f(x) = \begin{cases} a x^2 + 3 & \text{if } x < 2, \\ 2x + a & \text{if } x \geq 2. \end{cases} \] ### Step 1: Find the Left-Hand Limit as \( x \to 2^- \) For \( x < 2 \), the function is \( f(x) = a x^2 + 3 \). The left-hand limit as \( x \to 2^- \) is: \[ \lim_{x \to 2^-} f(x) = a(2)^2 + 3

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