For the function $f(x)=2 \sqrt{x-3}-2 \sqrt{2 x-4}$, find all values $x$ for which $f(x)=2$.
A) 2
B) 4
C) -2
D) $\varnothing$

Question

Quizplus · Accepted Answer

The equation $f(x) = 2$ leads to $2 \sqrt{x-3} - 2 \sqrt{2x-4} = 2$. Simplifying and solving this equation involves squaring both sides, which can introduce extraneous solutions. However, the key insight is that for all $x$ in the domain of $f$, $2 \sqrt{x-3} < 2 \sqrt{2x-4}$ because the function inside the square root is larger in the second term for all $x > 2$. This means the left side of the equation will always be less than 2, making it impossible for $f(x) = 2$. Therefore, there are no values of $x$ for which $f(x) = 2$.

For the Function f(x)=2x−3−22x−4f(x)=2 \sqrt{x-3}-2 \sqrt{2 x-4}f(x)=2x−3​−22x−4​ , Find All Values xxx

For the Function $f(x)=2 \sqrt{x-3}-2 \sqrt{2 x-4}$ , Find All Values $x$