Question

let $f(x)=\begin{cases}7x - 6&\text{if }xleq5\\-2x + b&\text{if }x > 5end{cases}$. if $f(x)$ is a function which is continuous everywhere, then we must have $b =$. now for fun, try to graph $f(x)$.

Explanation:

Step1: Recall continuity condition

For a function to be continuous at \(x = 5\), \(\lim_{x
ightarrow5^{-}}f(x)=\lim_{x
ightarrow5^{+}}f(x)\).

Step2: Calculate left - hand limit

\(\lim_{x
ightarrow5^{-}}f(x)\), since \(x
ightarrow5^{-}\) (approaching 5 from the left, \(x\leq5\)), \(f(x)=7x - 6\). So \(\lim_{x
ightarrow5^{-}}f(x)=7\times5 - 6=35 - 6 = 29\).

Step3: Calculate right - hand limit

\(\lim_{x
ightarrow5^{+}}f(x)\), since \(x
ightarrow5^{+}\) (approaching 5 from the right, \(x > 5\)), \(f(x)=-2x + b\). So \(\lim_{x
ightarrow5^{+}}f(x)=-2\times5 + b=-10 + b\).

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

Set \(-10 + b=29\). Solving for \(b\), we add 10 to both sides of the equation: \(b=29 + 10=39\).

Answer:

39