Question

Draw the graph of the function f, defined by f(x)= |x- 6| + |5 - x| ; x∈ [2,8]

Answer 1

$f(x)=|x-6|+|5-x|$

$|x-6|=(x-6), if (x-6)\geqslant0\iff x\geqslant6$

$|x-6|=-(x-6), if (x-6)<0\iff x<6$

$|5-x|=(5-x), if (5-x)\geqslant0\iff x\leqslant5$

$|5-x|=-(5-x), if (5-x)<0⟺x>5$

1) $x\in[2,5]$

$f(x)=-(x-6)+(5-x)=-2x+11$

2) $x\in(5, 6)$

$f(x)=-(x-6)-(5-x)=-x+6-5+x=1$

3) $x\in[6, 8]$

$f(x)=(x-6)-(5-x)=2x-11$

$f(x)=(x-6)-(5-x)=2x-11$