Question

find numbers a and b, or k, so that f is continuous at every point. 12) $f(x)=\begin{cases}x^{2},& \text{if }xleq4\\x + k,&\text{if }x>4end{cases}$ a) $k = 12$ b) $k = 20$

Explanation:

Step1: Recall continuity condition

For a function to be continuous at \(x = 4\), \(\lim_{x
ightarrow4^{-}}f(x)=\lim_{x
ightarrow4^{+}}f(x)\).

Step2: Calculate left - hand limit

\(\lim_{x
ightarrow4^{-}}f(x)=\lim_{x
ightarrow4^{-}}x^{2}\). Substitute \(x = 4\) into \(x^{2}\), we get \(4^{2}=16\).

Step3: Calculate right - hand limit

\(\lim_{x
ightarrow4^{+}}f(x)=\lim_{x
ightarrow4^{+}}(x + k)\). Substitute \(x = 4\) into \(x + k\), we get \(4 + k\).

Step4: Set left - hand and right - hand limits equal

Set \(4 + k=16\). Solve for \(k\): \(k=16 - 4=12\).

Answer:

A. \(k = 12\)