How do you write #y = |6 + 2x| + 1 # as a piecewise function?

Question

Isaiah Anthony · Accepted Answer

#y=-(6+2x)+1;x <-3# and
#y=(6+2x)+1;x >-3# We could say that the absolute equation is shifted 1 units up but in the same x axis. So, We set the expression inside the bar to zero because the absolute means distance from zero but not less than zero. Hence, #abs (6+2x)=0# #Or, 6=-2x# Dividing both sides by #-2# #6/-2=x# #:.x=-3# Hence, the graph will bounce off at #x=-3# Which divides the graph into two intervals. #(-oo,-3) and (-3,oo)#. Note that we aren't including #-3# because it will simply turn the expression inside the bar to be zero. Hence our final piecewise function is. #y=-(6+2x)+1; if x <-3# #y=(6+2x)+1; if x>-3# Note that we are excluding the vertical shift of #1# units because it doesn't afflict the equation.

Hazel Anglin · Answer

undefined To write $ y = |6 + 2x| + 1 $ as a piecewise function, we can break it down based on the sign of the expression inside the absolute value:

$$ y = \begin{cases} 6 + 2x + 1 & \text{if } 6 + 2x \geq 0 \ -(6 + 2x) + 1 & \text{if } 6 + 2x < 0 \end{cases} $$

Simplifying further:

$$ y = \begin{cases} 2x + 7 & \text{if } x \geq -3 \ -2x + 7 & \text{if } x < -3 \end{cases} $$