Question

Rewrite as a piece wise function y=1/2 |x-6| +4

Answer 1

y = {-1/2x +7 for x < 6; 1/2x +1 for x ≥ 6}

Explanation:

You want y = 1/2|x -6| +4 written as a piecewise function.

Domains

The absolute value function changes its definition when its argument is negative:

y = |x| ⇒ y = -x for x < 0, and y = x for x ≥ 0

This means our piecewise function will have one definition for (x-6) < 0 and another for (x -6) ≥ 0.

For x -6 < 0

The argument is negated in this domain, so we have ...

y = -1/2(x -6) +4

y = -1/2x +3 +4

y = -1/2x +7

For x -6 ≥ 0

The absolute value function is an identity function in this domain:

y = 1/2(x -6) +4

y = 1/2x -3 +4

y = 1/2x +1

Piecewise function

Combining these descriptions into one, we have ...

`\boxed{y=\begin{cases}-(1)/(2)x+7&\text{for }x < 6\\\\(1)/(2)x+1&\text{for }x\ge6\end{cases}}`

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