To find the inverse of the function [tex]\( y = x^2 - 10x \)[/tex], follow these steps:
1. Rewrite the original function:
[tex]\[ y = x^2 - 10x \][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = y^2 - 10y \][/tex]
3. Reformat the equation to zero on one side:
[tex]\[ y^2 - 10y - x = 0 \][/tex]
4. Consider solving for [tex]\( y \)[/tex]. This is a quadratic equation in terms of [tex]\( y \)[/tex]:
[tex]\[ y^2 - 10y - x = 0 \][/tex]
5. Use the quadratic formula [tex]\( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -10 \)[/tex], and [tex]\( c = -x \)[/tex]:
[tex]\[ y = \frac{10 \pm \sqrt{(-10)^2 - 4(1)(-x)}}{2(1)} \][/tex]
[tex]\[ y = \frac{10 \pm \sqrt{100 + 4x}}{2} \][/tex]
[tex]\[ y = \frac{10 \pm \sqrt{4(x + 25)}}{2} \][/tex]
[tex]\[ y = \frac{10 \pm 2\sqrt{x + 25}}{2} \][/tex]
[tex]\[ y = 5 \pm \sqrt{x + 25} \][/tex]
6. Thus, there are two possible solutions for the inverse function:
[tex]\[ y = 5 + \sqrt{x + 25} \][/tex]
[tex]\[ y = 5 - \sqrt{x + 25} \][/tex]
Upon simplifying, the inverse functions are:
[tex]\[ y = \sqrt{x + 25} + 5 \][/tex]
and
[tex]\[ y = \sqrt{x + 25} - 5 \][/tex]
Given these two forms, we compare the given choices:
- [tex]\( y = \pm \sqrt{x-25}-5 \)[/tex]
- [tex]\( y = \pm \sqrt{x-25}+5 \)[/tex]
- [tex]\( y = \pm \sqrt{x+25}-5 \)[/tex]
- [tex]\( y = \pm \sqrt{x+25}+5 \)[/tex]
The correct forms derived are:
[tex]\[ y = \pm \sqrt{x + 25} - 5 \][/tex]
and
[tex]\[ y = \pm \sqrt{x + 25} + 5 \][/tex]
So, the correct answer is:
[tex]\[ y = \pm \sqrt{x + 25} - 5 \][/tex]
