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Consider the function f(x)=|-x-3/2|-1
and
The piecewise functions that coincide with the given function are

Consider The Function Fxx321 And The Piecewise Functions That Coincide With The Given Function Are class=

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LammettHashid

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

What is piecewise function?

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:

https://brainly.com/question/12561612

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