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Let f(x) be defined by

f(x) = x - 3 if x <= 3

f(x) = x^2 - 6x + 9 if x > 3

 

Calculate f^{-1}(0) + f^{-1}(6).

 May 15, 2022
 #1
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Note that the function is increasing everywhere (so when we find a solution to f(x) = k, that must be the unique solution).

When f(x) = 0, x = 3 since f(3) = 3 - 3 = 0 (solution is unique because f is increasing).

That means \(f^{-1}(0) = 3\).

For any x <= 3, f(x) = x - 3 <= 0.

Therefore, when f(x) = 6, x > 3.

Can you try to find the value of x from here? That value would be \(f^{-1}(6)\), and now you just add the results together.

 May 15, 2022

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