Question

How do you write `y= |x-5| -4` as a piecewise function?

Answer 1

Start with the piecewise definition of the absolute value function:

`|f(x)| = {(f(x); f(x) >= 0),(-f(x); f(x) < 0):}`

Substitute `x - 5` for `f(x)`

`|x - 5| = {(x - 5; x - 5 >= 0),(-(x - 5); x - 5 < 0):}`

Simplify the inequalities:

`|x - 5| = {(x - 5; x >= 5),(-(x - 5); x < 5):}`

Distribute the -1:

`|x - 5| = {(x - 5; x >= 5),(5-x; x < 5):}`

Make two separate equation of the original:

`y = {((x-5)-4;x>=5),((5-x)-4; x < 5):}`

Simplify:

`y = {(x-9;x>=5),(1-x; x < 5):}`

Answer 2

`y=-x+1` for `x<=5`
`y=x-9` for `x>5`

Explanation:

First let's look at the graph or the original function `y=abs(x-5)`


The left half of the graph where `x<=5` has a slope of `m=-1` and a y intercept of `(0,1)`.

Using the slope intercept form of a line `y=mx+b` where `m=-1` and `b=1` gives the equation is `y=-1x+1` or `y=-x+1`

The right half of the graph where `x>=5` has a slope of `m=1` and passes through the point `(5,-4)`, which is the vertex of the graph as shown in the original equation.

Using the slope intercept form of the line `y-y_1=m(x-x_1)` where `m=1` and `(x_1,y_1) = (5,-4)` gives the equation

`y- -4 = 1 (x-5)`
`y+4= x-5`
`y=x-9` for `x<=5`

Written in piecewise form

`y=-x+1` for `x<=5`
`y=x-9` for `x>5`

Note that I arbitrarily assigned `x=5` to the first equation.