Question

How do you write `H(x) = 2/3|x-5|` as a piecewise function?

Answer 1

`H (x) = -2/3 (x-5), x <5`
`H (x) = 2/3 (x-5) , x >=5`

Explanation:

If x is less than 5, the number within the absolute value will be negative. Thus, if we wish to rid ourselves of the absolute value symbol, which as one recalls changes any negative expression within it positive, we must find a way to make `2/3 (x-5)` positive when `x <5`.

This is most easily accomplished by multiplying the expression by -1 when `x <5` ; in this case, since it will only be applied when `2/3 (x-5) <0`, it will turn the expression positive. The breakpoint is a at x=5, because at x=5 the expression is equal to 0, and as x increases the expression becomes more positive.

Thus, we are left with
`H (x) = -2/3 (x-5), x <5`
`H (x) = 2/3 (x-5) , x >=5`