Question

If the function \[ f(x) = \begin{cases} x^2, & \text{for } x < 4 \\ 5x - k, & \text{for } x \geq 4 \end{cases} \] is continuous at \( x = 4 \), then the value of \( k \) is equal to

• A. \( 2 \)
• B. \( 3 \)
• C. \( 4 \)
• D. \( 5 \)
• E. \( 6 \)

Hint:

For a function to be continuous at \( x = a \), ensure that \( \lim\limits_{x \to a^-} f(x) = \lim\limits_{x \to a^+} f(x) = f(a) \).

Answer 1

The Correct Option is C

For continuity at \( x = 4 \), \[ \lim_{x \to 4^-} f(x) = \lim_{x \to 4^+} f(x) = f(4). \] From the left-hand limit, \[ \lim_{x \to 4^-} f(x) = 4^2 = 16. \] From the right-hand limit, \[ \lim_{x \to 4^+} f(x) = 5(4) - k = 20 - k. \] Equating both sides, \[ 16 = 20 - k. \] Solving for \( k \), \[ k = 4. \] Thus, the correct answer is (C).