The inverse of a function, is its opposite.
The inverse function is: [tex]\mathbf{f^{-1}(x) = 5 \pm \sqrt{x+25} }[/tex]
The function is given as:
[tex]\mathbf{y = x^2 - 10x}[/tex]
Swap y and x
[tex]\mathbf{x = y^2 - 10y}[/tex]
Complete the square on the above equation
[tex]\mathbf{x+ (-\frac{10}{2})^2 = y^2 - 10y + (-\frac{10}{2})^2}[/tex]
[tex]\mathbf{x+25 = y^2 - 10y + 25}[/tex]
Express the right-hand side as a perfect square
[tex]\mathbf{x+25 = (y - 5)^2}[/tex]
Take square roots of both sides
[tex]\mathbf{\pm \sqrt{x+25} = y - 5}[/tex]
Add 5 to both sides
[tex]\mathbf{5 \pm \sqrt{x+25} = y }[/tex]
Rewrite as:
[tex]\mathbf{y = 5 \pm \sqrt{x+25} }[/tex]
So, we have:
[tex]\mathbf{f^{-1}(x) = 5 \pm \sqrt{x+25} }[/tex]
Hence, the inverse function is: [tex]\mathbf{f^{-1}(x) = 5 \pm \sqrt{x+25} }[/tex]
Read more about inverse functions at:
https://brainly.com/question/10300045
