Recall the definition of absolute value,
[tex]|x| = \begin{cases}x&\text{if }x\ge0\\-x&\text{if }x<0\end{cases}[/tex]
• If (-x - 3)/2 ≥ 0, then |(-x - 3)/2| = (-x - 3)/2. So
f(x) = (-x - 3)/2 - 1 = (-x - 5)/2
Simplifying the condition gives
(-x - 3)/2 ≥ 0 ⇒ -x - 3 ≥ 0 ⇒ x ≤ -3
• Otherwise, if (-x - 3)/2 < 0, then |(-x - 3)/2| = -(-x - 3)/2, and we have
f(x) = -(-x - 3)/2 - 1 = (x + 3)/2 - 1 = (x + 1)/2
and
(-x - 3)/2 < 0 ⇒ -x - 3 < 0 ⇒ x > -3
So, as a piecewise function, we can write
[tex]f(x) = \left|\dfrac{-x-3}2\right| - 1 = \begin{cases}\dfrac{-x-5}2&\text{if }x\le-3\\\\\dfrac{x+1}2&\text{if }x>-3\end{cases}[/tex]
The piecewise functions that coincide with the given function is
f(x) = [tex]\frac{-(x+3)}{2} -1[/tex] , where x is greater than or equal to -3
A piecewise-defined function is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. Piecewise definition is actually a way of expressing the function, rather than a characteristic of the function itself.
According to the question
f(x) = [tex]|\frac{-x-3}{2} |-1[/tex]
The piecewise functions are :
f(x) = [tex]\frac{-(x+3)}{2} -1[/tex]
Case 1:
when x ≥ -3
F(x) = [tex]\frac{-(x+3)}{2} -1[/tex]
Case2:
when x < -3
F(x) = [tex]\frac{(x+3)}{2} -1[/tex]
Hence, The piecewise functions that coincide with the given function is
f(x) = [tex]\frac{-(x+3)}{2} -1[/tex] , where x is greater than or equal to -3
To know more about piecewise function here:
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