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Sagot :
To determine which equation is the inverse of [tex]\( y = 100 - x^2 \)[/tex], we need to solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex] from the given equation.
1. Start with the original equation:
[tex]\[ y = 100 - x^2 \][/tex]
2. To find the inverse, solve for [tex]\( x \)[/tex]:
[tex]\[ y = 100 - x^2 \][/tex]
[tex]\[ x^2 = 100 - y \][/tex]
[tex]\[ x = \pm \sqrt{100 - y} \][/tex]
The inverse relationship is [tex]\( x = \pm \sqrt{100 - y} \)[/tex]. However, we need to express this in the form of [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].
Rewriting the inverse equation:
[tex]\[ y = 100 - x^2 \quad \Rightarrow \quad x = \pm \sqrt{100 - y} \][/tex]
Since we need the form where [tex]\( y \)[/tex] is the subject in all choices, we can reject certain choices.
Now, consider the choices given:
1. [tex]\( y = \pm \sqrt{100 - x} \)[/tex]
2. [tex]\( y = 10 \pm \sqrt{x} \)[/tex]
3. [tex]\( y = 100 \pm \sqrt{x} \)[/tex]
4. [tex]\( y = \pm \sqrt{x - 100} \)[/tex]
Based on the inverse relationship [tex]\( x = \pm \sqrt{100 - y} \)[/tex], we know it should be expressed as [tex]\( y \)[/tex] being related to [tex]\( 100 - x \)[/tex], since we originally solved for [tex]\( x \)[/tex]:
Analyzing the choices:
1. [tex]\( y = \pm \sqrt{100 - x} \)[/tex] — In this choice, the structure matches [tex]\( x = \pm \sqrt{100 - y} \)[/tex], however, it's not in the form expected from our inverse.
2. [tex]\( y = 10 \pm \sqrt{x} \)[/tex] — This form does not match our required relationship.
3. [tex]\( y = 100 \pm \sqrt{x} \)[/tex] — This also doesn’t match our specified inverse structure.
4. [tex]\( y = \pm \sqrt{x - 100} \)[/tex] — This is not related to our derived inverse.
Thus, there isn't a direct matching form among the given choices that correctly represents the inverse equation we derived.
Since none of the given equations properly match the inverse of [tex]\( y = 100 - x^2 \)[/tex], the answer is:
None of these choices provide the correct inverse.
1. Start with the original equation:
[tex]\[ y = 100 - x^2 \][/tex]
2. To find the inverse, solve for [tex]\( x \)[/tex]:
[tex]\[ y = 100 - x^2 \][/tex]
[tex]\[ x^2 = 100 - y \][/tex]
[tex]\[ x = \pm \sqrt{100 - y} \][/tex]
The inverse relationship is [tex]\( x = \pm \sqrt{100 - y} \)[/tex]. However, we need to express this in the form of [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].
Rewriting the inverse equation:
[tex]\[ y = 100 - x^2 \quad \Rightarrow \quad x = \pm \sqrt{100 - y} \][/tex]
Since we need the form where [tex]\( y \)[/tex] is the subject in all choices, we can reject certain choices.
Now, consider the choices given:
1. [tex]\( y = \pm \sqrt{100 - x} \)[/tex]
2. [tex]\( y = 10 \pm \sqrt{x} \)[/tex]
3. [tex]\( y = 100 \pm \sqrt{x} \)[/tex]
4. [tex]\( y = \pm \sqrt{x - 100} \)[/tex]
Based on the inverse relationship [tex]\( x = \pm \sqrt{100 - y} \)[/tex], we know it should be expressed as [tex]\( y \)[/tex] being related to [tex]\( 100 - x \)[/tex], since we originally solved for [tex]\( x \)[/tex]:
Analyzing the choices:
1. [tex]\( y = \pm \sqrt{100 - x} \)[/tex] — In this choice, the structure matches [tex]\( x = \pm \sqrt{100 - y} \)[/tex], however, it's not in the form expected from our inverse.
2. [tex]\( y = 10 \pm \sqrt{x} \)[/tex] — This form does not match our required relationship.
3. [tex]\( y = 100 \pm \sqrt{x} \)[/tex] — This also doesn’t match our specified inverse structure.
4. [tex]\( y = \pm \sqrt{x - 100} \)[/tex] — This is not related to our derived inverse.
Thus, there isn't a direct matching form among the given choices that correctly represents the inverse equation we derived.
Since none of the given equations properly match the inverse of [tex]\( y = 100 - x^2 \)[/tex], the answer is:
None of these choices provide the correct inverse.
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