Answer
$\{6\}$.
Work Step by Step
The given function is
$f(x)=x-\sqrt{x-2}$
Replace $f(x)$ with $4$.
$\Rightarrow 4=x-\sqrt{x-2}$
Add $\sqrt{x-2}-4$ to both sides.
$\Rightarrow 4+\sqrt{x-2}-4=x-\sqrt{x-2}+\sqrt{x-2}-4$
Simplify.
$\Rightarrow \sqrt{x-2}=x-4$
Square both sides.
$\Rightarrow (\sqrt{x-2})^2=(x-4)^2$
Use the special formula $(A-B)^2=A^2-2AB+B^2$
We have $A=x$ and $B=4$
$\Rightarrow x-2=(x)^2-2(x)( 4)+(4)^2$
Simplify.
$\Rightarrow x-2=x^2-8x+16$
Add $-x+2$ to both sides.
$\Rightarrow x-2-x+2=x^2-8x+16-x+2$
Add like terms.
$\Rightarrow 0=x^2-9x+18$
Rewrite the middle term $-9x$ as $-6x-3x$.
$\Rightarrow x^2-6x-3x+18=0$
Group terms.
$\Rightarrow (x^2-6x)+(-3x+18)=0$
Factor each group.
$\Rightarrow x(x-6)-3(x-6)=0$
Factor out $(x-6)$.
$\Rightarrow (x-6)(x-3)=0$
Set each factor equal to zero.
$\Rightarrow x-6=0$ or $ x-3=0$
Isolate $x$.
$\Rightarrow x=6$ or $ x=3$
Check $x=6$.
$f(6)=6-\sqrt{6-2}$
$f(6)=6-\sqrt{4}$
$f(6)=6-2$
$f(6)=4$ true.
Check $x=3$.
$f(3)=3-\sqrt{3-2}$
$f(3)=3-\sqrt{1}$
$f(3)=3-1$
$f(3)=2$ false.
The solution set is $\{6\}$.
