Questions: In the following exercises, find the value(s) of (k) that makes each function continuous over the given interval. (f(x)=beginarrayll3 x+2, x<k 2 x-3, k leq x leq 8endarray)

Solution Steps

To find the value of \( k \) that makes the function \( f(x) \) continuous over the given interval, we need to ensure that the left-hand limit and the right-hand limit at \( x = k \) are equal. This means solving for \( k \) such that \( 3k + 2 = 2k - 3 \).

We are given the piecewise function: \[ f(x) = \begin{cases} 3x + 2, & \text{if } x < k \\ 2x - 3, & \text{if } k \leq x \leq 8 \end{cases} \] We need to find the value of \( k \) that makes \( f(x) \) continuous over the given interval.

To ensure continuity at \( x = k \), the left-hand limit and the right-hand limit at \( x = k \) must be equal: \[ \lim_{{x \to k^-}} f(x) = \lim_{{x \to k^+}} f(x) \] This translates to: \[ 3k + 2 = 2k - 3 \]

Solving the equation \( 3k + 2 = 2k - 3 \): \[ 3k + 2 = 2k - 3 \] Subtract \( 2k \) from both sides: \[ k + 2 = -3 \] Subtract 2 from both sides: \[ k = -5 \]

Final Answer