To find the inverse of the given equation [tex]\( y = 100 - x^2 \)[/tex], follow these steps:
1. Start with the original equation:
[tex]\[
y = 100 - x^2
\][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse:
[tex]\[
x = 100 - y^2
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
[tex]\[
y^2 = 100 - x
\][/tex]
To isolate [tex]\( y \)[/tex], take the square root of both sides:
[tex]\[
y = \pm \sqrt{100 - x}
\][/tex]
Therefore, the equation that represents the inverse of [tex]\( y = 100 - x^2 \)[/tex] is:
[tex]\[
y = \pm \sqrt{100 - x}
\][/tex]
Now, let's look at the provided answer choices and find the correct one:
1. [tex]\( y = \pm \sqrt{100 - x} \)[/tex]
2. [tex]\( y = 10 \pm \sqrt{x} \)[/tex]
3. [tex]\( y = 100 = \sqrt{x} \)[/tex]
4. [tex]\( y = \pm \sqrt{x - 100} \)[/tex]
The correct answer, based on our step-by-step solution, is:
[tex]\[
y = \pm \sqrt{100 - x}
\][/tex]
So the correct choice is the first option:
[tex]\[ y = \pm \sqrt{100 - x} \][/tex]
