Question

Consider the function
$f\left(x\right)=\{\begin{array}{c}2x-1,0\le x<2\\ x+a2\le x\le 4\\ 3x+b4<x\le 6\end{array}$
(i) Find $f(2-)$ and $f(2+)$
(ii) Find $a$ if $f$ is continuous at $x=2$
(iii) Find $b$ if $f$ is continuous on $[0,6]$

Answer 1

$f(x<2)=2x-1$

$\Rightarrow f(2-)=2\left(2\right)-1=3$

$f(x>2)=x+a$

$\Rightarrow f(2+)=2+a$

Given that $f$ is continous at $x=2$

$\Rightarrow f(2-)=f(2+)$

$\Rightarrow 2+a=3$

$\Rightarrow a=-1$

Given that $f$ is continous on $[0,6]$

$\Rightarrow f(4-)=f(4+)$

$\Rightarrow 4+a=12+b$

$\Rightarrow 4-1=12+b$

$\Rightarrow b=-9$