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Let

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

f(x) = 3x + 1 if x < -4

 

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

 Apr 5, 2022
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f(f(x)) = 5 = f(x')

If x' >= -4, then 5 = x'^2 - 4; x' = 3.

If x' < -4, then 5 = 3x' + 1; x' = 4/3.

 

Case 1: f(x) = 3

If x >= -4, then 3 = x^2 - 4; x = sqrt(7).

If x < -4, then 3 = 3x + 1; x = 2/3.

Case 2: f(x) = 4/3

If x >= -4, then 4/3 = x^2 - 4; x = 4sqrt(3)/3.

If x < -4, then 4/3 = 3x + 1; x = 1/9.

 

Thus there are 4 values of x that work, and the values are above.

 

There is a much quicker way to do it than calculating it; try looking at how many options there are for the range of x (in this case there are 2: x >= -4, x < -4) then multiply by the #of 'f's (in this case 2: f(f(x))). 2 x 2 = 4. :)

 Apr 5, 2022

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