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  4. Let f (x)=x2-x+1, x ge(1/2), then the solution of the equation f(x) = f-1(x) is

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Q. Let $f \left(x\right)=x^{2}-x+1$, $x \ge\frac{1}{2}$, then the solution of the equation $f(x) = f^{-1}(x)$ is

Relations and Functions - Part 2

Solution:

Let $y=x^{2}-x+1$
$\Rightarrow x=\frac{1 \pm\sqrt{1-4\left(1-y\right)}}{2}\because x \ge\frac{1}{2}$
$\therefore x=\frac{1}{2}+\sqrt{y-\frac{3}{4}}$
$\Rightarrow f ^{-1}\left(x\right)=\frac{1}{2}+\sqrt{x-\frac{3}{4}}$
Now, $x^{2}-x+1=\frac{1}{2}+\sqrt{x-\frac{3}{4}}$
$\Rightarrow x=1$ (By checking the given possibilities)