Question

How do you write `y = |9 - x| +7` into piecewise functions?

Answer 1

`y=16-x, if 9>=x,` and, `y=x-2, x>9.`

Explanation:

Recall that `AA t in RR, |t|=t. if t>=0, and, |t|=-t, if t<0.`

Accordingly, `(i) |9-x|=9-x, if (9-x)>=0, i.e., 9>=x,` and,
`(ii) |9-x|=-(9-x), if (9-x)<0. or, x>9.`

Therefore, in `(i), y=|9-x|+7=9-x+7=16-x, if, 9>=x`, whereas,
in `(ii), y=-(9-x)=7=-9+x+7=x-2, if, x>9.`

Altogether, `y=16-x, if 9>=x,` and, `y=x-2, x>9.`