If $f(x) = x^2 - 2ax + a(a+1)$ , $f:[ a, \infty] \to [a,\infty]$ . If one of the solution of the equation $f(x)=f^{-1}(x)$ is $5049$ , then what may be the other solution ?
My WORK:
I found the inverse, but when I equate them , it is very difficult to solve . Please help me with some method to solve it.
If $t = f^{-1}(x)$, you can write this as $f(f(t)) = t$, and this becomes $$ (t-a)^4 = t-a$$ So either $t=a$, $x = a$, or $(t-a)^3 = 1$, $t = a + 1$, $x = a+1$.
Thus either $a = 5049$ and the other solution is $a+1=5050$, or $a+1 = 5049$ and the other solution is $a = 5048$.