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Answer :
To find the correct piecewise definition for the function [tex]\( y = |x + 5| - 2 \)[/tex], we need to break it down into two parts based on the expression inside the absolute value.
1. Identify the critical point:
The critical point is found when the expression inside the absolute value is zero. Here's how we do it:
[tex]\[
x + 5 = 0 \implies x = -5
\][/tex]
This critical point divides the function into two parts - one for [tex]\( x \geq -5 \)[/tex] and another for [tex]\( x < -5 \)[/tex].
2. Define the function for [tex]\( x \geq -5 \)[/tex]:
For [tex]\( x \geq -5 \)[/tex], the absolute value expression [tex]\( |x + 5| \)[/tex] simplifies to [tex]\( x + 5 \)[/tex], since positive values or zero don't change when the absolute value is removed. So the function becomes:
[tex]\[
y = (x + 5) - 2 = x + 3
\][/tex]
Therefore, on this interval, the function is [tex]\( y = x + 3 \)[/tex].
3. Define the function for [tex]\( x < -5 \)[/tex]:
For [tex]\( x < -5 \)[/tex], the absolute value expression [tex]\( |x + 5| \)[/tex] needs to be considered as negative to handle when [tex]\( x + 5 \)[/tex] is negative, flipping its sign. Thus, the function becomes:
[tex]\[
y = -(x + 5) - 2 = -x - 5 - 2 = -x - 7
\][/tex]
Therefore, on this interval, the function is [tex]\( y = -x - 7 \)[/tex].
Putting these together, the piecewise definition of the function is:
- [tex]\( y = x + 3 \)[/tex] for [tex]\( x \geq -5 \)[/tex]
- [tex]\( y = -x - 7 \)[/tex] for [tex]\( x < -5 \)[/tex]
Thus, the correct piecewise definition for the function [tex]\( y = |x+5|-2 \)[/tex] is:
[tex]\[ y = x + 3 \text{ for } x \geq -5 \text{ and } y = -x - 7 \text{ for } x < -5. \][/tex]
1. Identify the critical point:
The critical point is found when the expression inside the absolute value is zero. Here's how we do it:
[tex]\[
x + 5 = 0 \implies x = -5
\][/tex]
This critical point divides the function into two parts - one for [tex]\( x \geq -5 \)[/tex] and another for [tex]\( x < -5 \)[/tex].
2. Define the function for [tex]\( x \geq -5 \)[/tex]:
For [tex]\( x \geq -5 \)[/tex], the absolute value expression [tex]\( |x + 5| \)[/tex] simplifies to [tex]\( x + 5 \)[/tex], since positive values or zero don't change when the absolute value is removed. So the function becomes:
[tex]\[
y = (x + 5) - 2 = x + 3
\][/tex]
Therefore, on this interval, the function is [tex]\( y = x + 3 \)[/tex].
3. Define the function for [tex]\( x < -5 \)[/tex]:
For [tex]\( x < -5 \)[/tex], the absolute value expression [tex]\( |x + 5| \)[/tex] needs to be considered as negative to handle when [tex]\( x + 5 \)[/tex] is negative, flipping its sign. Thus, the function becomes:
[tex]\[
y = -(x + 5) - 2 = -x - 5 - 2 = -x - 7
\][/tex]
Therefore, on this interval, the function is [tex]\( y = -x - 7 \)[/tex].
Putting these together, the piecewise definition of the function is:
- [tex]\( y = x + 3 \)[/tex] for [tex]\( x \geq -5 \)[/tex]
- [tex]\( y = -x - 7 \)[/tex] for [tex]\( x < -5 \)[/tex]
Thus, the correct piecewise definition for the function [tex]\( y = |x+5|-2 \)[/tex] is:
[tex]\[ y = x + 3 \text{ for } x \geq -5 \text{ and } y = -x - 7 \text{ for } x < -5. \][/tex]
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