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The function f satisfies
f(\sqrt{2x} + x) = \frac{1}{x}
for all x \ge \frac{1}{2}.  Find f(4).

 Feb 16, 2024

Best Answer 

 #1
avatar+1633 
+2

The function f satisfies f(2x+x)=1x for all x12, find f(4).

f(4)=f(2x+x)=1x

2x+x=42x=x+42x=x28x+16x210x+16=0(x8)(x2)=0

Note that x = 8 i s an extraneous solution, because 28 does not equal to 8+4=4, for a square root can't end up negative.

Thus, our only solution, x = 2. f(2(2)+2)=12=f(4), so our final answer is 1/2.

 Feb 16, 2024
 #1
avatar+1633 
+2
Best Answer

The function f satisfies f(2x+x)=1x for all x12, find f(4).

f(4)=f(2x+x)=1x

2x+x=42x=x+42x=x28x+16x210x+16=0(x8)(x2)=0

Note that x = 8 i s an extraneous solution, because 28 does not equal to 8+4=4, for a square root can't end up negative.

Thus, our only solution, x = 2. f(2(2)+2)=12=f(4), so our final answer is 1/2.

proyaop Feb 16, 2024

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