To find the inverse of the function [tex]\( y = 100 - x^2 \)[/tex], we need to swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] and then solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]. Here's the detailed step-by-step process:
1. Start with the original function:
[tex]\[
y = 100 - x^2
\][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x = 100 - y^2
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
[tex]\[
y^2 = 100 - x
\][/tex]
[tex]\[
y = \pm\sqrt{100 - x}
\][/tex]
Therefore, the inverse function is:
[tex]\[
y = \pm\sqrt{100 - x}
\][/tex]
By comparing with the given options:
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]
We see that the correct inverse function is:
[tex]\[ y = \pm \sqrt{100 - x} \][/tex]
Thus, the correct answer is:
[tex]\[ y = \pm \sqrt{100 - x} \][/tex]
