b To write the absolute value function as a piecewise function, we first need to determine where the function changes direction as this is where our domains will split. In this case, this occurs when x=5.
When we find the domain for the left piece we keep in mind that the function begins at (3,0).
DomainLeftPiece:3≤x<5
The possible values for the right piece of the function are all real numbers greater than or equal to 5.
DomainRightPiece:x≤5
Next, we can write the function as two separate equations by splitting our absolute value into its two cases, one positive and one negative.
y={-2[-(x−5)]+4,-2(x−5)+4,3≤x<5x≥5
We can simplify these equations a little bit. Let's start with the negative case.
Finally, we can write the equation of the piecewise function.
y={2x−6,-2x+14,3≤x<5x≥5
Extra
Alternative Method
Rather than using the absolute value equation to solve for the piecewise function, as the example in the textbook showed, we can write the equations using the slope-intercept form.
y=mx+b
We can use the graph of the absolute value function to find the slope of the left piece and the slope of the right piece. Then we use points from the graph, such as (5,4), to solve for b. If done correctly, the answer will be the same.
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