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Answer :
To find the correct piecewise definition for the function [tex]\( y = |x + 5| - 2 \)[/tex], we can analyze how the absolute value function works and then break it down based on the piecewise conditions.
1. Identify the critical point:
- The critical point of the expression inside the absolute value, [tex]\( |x + 5| \)[/tex], is where [tex]\( x + 5 = 0 \)[/tex]. Solving for [tex]\( x \)[/tex], we have [tex]\( x = -5 \)[/tex].
- At [tex]\( x = -5 \)[/tex], the expression inside changes from negative to non-negative.
2. Evaluate for [tex]\( x < -5 \)[/tex]:
- For values of [tex]\( x \)[/tex] less than [tex]\(-5\)[/tex], the expression [tex]\( x + 5 \)[/tex] is negative. Therefore, [tex]\( |x + 5| = -(x + 5) \)[/tex].
- Substituting back into the original function:
[tex]\[
y = |x + 5| - 2 = -(x + 5) - 2 = -x - 5 - 2 = -x - 7
\][/tex]
- So, the piece for [tex]\( x < -5 \)[/tex] is [tex]\( y = -x - 7 \)[/tex].
3. Evaluate for [tex]\( x \geq -5 \)[/tex]:
- For values of [tex]\( x \)[/tex] greater than or equal to [tex]\(-5\)[/tex], the expression [tex]\( x + 5 \)[/tex] is non-negative. Thus, [tex]\( |x + 5| = x + 5 \)[/tex].
- Substituting back into the original function:
[tex]\[
y = |x + 5| - 2 = x + 5 - 2 = x + 3
\][/tex]
- So, the piece for [tex]\( x \geq -5 \)[/tex] is [tex]\( y = x + 3 \)[/tex].
Now putting both pieces together, the correct piecewise definition for the function is:
- [tex]\( y = -x - 7 \)[/tex] for [tex]\( x < -5 \)[/tex]
- [tex]\( y = x + 3 \)[/tex] for [tex]\( x \geq -5 \)[/tex]
Comparing these with the given options, the correct choice is:
- [tex]\( y = x + 3 \)[/tex] for [tex]\( x \geq -5 \)[/tex] and [tex]\( y = -x - 7 \)[/tex] for [tex]\( x < -5 \)[/tex].
This corresponds to option 2 in the list of provided choices.
1. Identify the critical point:
- The critical point of the expression inside the absolute value, [tex]\( |x + 5| \)[/tex], is where [tex]\( x + 5 = 0 \)[/tex]. Solving for [tex]\( x \)[/tex], we have [tex]\( x = -5 \)[/tex].
- At [tex]\( x = -5 \)[/tex], the expression inside changes from negative to non-negative.
2. Evaluate for [tex]\( x < -5 \)[/tex]:
- For values of [tex]\( x \)[/tex] less than [tex]\(-5\)[/tex], the expression [tex]\( x + 5 \)[/tex] is negative. Therefore, [tex]\( |x + 5| = -(x + 5) \)[/tex].
- Substituting back into the original function:
[tex]\[
y = |x + 5| - 2 = -(x + 5) - 2 = -x - 5 - 2 = -x - 7
\][/tex]
- So, the piece for [tex]\( x < -5 \)[/tex] is [tex]\( y = -x - 7 \)[/tex].
3. Evaluate for [tex]\( x \geq -5 \)[/tex]:
- For values of [tex]\( x \)[/tex] greater than or equal to [tex]\(-5\)[/tex], the expression [tex]\( x + 5 \)[/tex] is non-negative. Thus, [tex]\( |x + 5| = x + 5 \)[/tex].
- Substituting back into the original function:
[tex]\[
y = |x + 5| - 2 = x + 5 - 2 = x + 3
\][/tex]
- So, the piece for [tex]\( x \geq -5 \)[/tex] is [tex]\( y = x + 3 \)[/tex].
Now putting both pieces together, the correct piecewise definition for the function is:
- [tex]\( y = -x - 7 \)[/tex] for [tex]\( x < -5 \)[/tex]
- [tex]\( y = x + 3 \)[/tex] for [tex]\( x \geq -5 \)[/tex]
Comparing these with the given options, the correct choice is:
- [tex]\( y = x + 3 \)[/tex] for [tex]\( x \geq -5 \)[/tex] and [tex]\( y = -x - 7 \)[/tex] for [tex]\( x < -5 \)[/tex].
This corresponds to option 2 in the list of provided choices.
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