Which equation is the inverse of [tex]y = 100 - x^2[/tex]?

A. [tex]y = \pm \sqrt{100 - x}[/tex]
B. [tex]y = 10 \pm \sqrt{x}[/tex]
C. [tex]y = 100 \pm \sqrt{x}[/tex]
D. [tex]y = \pm \sqrt{x - 100}[/tex]



Answer :

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.