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If

f(x) = x^2 - 4 if x >= 4

f(x) = x^2 + 4 otherwise,

then for how many values of x is f(f(x)) = 5?

 May 3, 2022
 #1
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When f(x)4(x244 and x4) or (x2+44 and x<4), which is x<4 or x4, which means f(x) is always greater than or equal to 4.

 

Then f(f(x)) = (f(x))^2 - 4 by definition of f.

 

Case 1: x >= 4

(x24)24=5x48x2+1645=0x48x2+7=0(x27)(x21)=0x=±7 (rej.) or x=±1 (rej.)

The roots are rejected since none of them are greater than 4.

 

Case 2: x < 4

(x2+4)24=5x4+8x2+1645=0(x2+1)(x2+7)=0No real solution

 

Therefore, there are no value of x that would make f(f(x)) = 5.

 May 3, 2022

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