Sum of Inverses

Algebra Level 3

Suppose that function f ( x ) f(x) is defined as f ( x ) = { x + 3 if x < 20 , 2 x 2 if x 20. f(x)= \begin{cases} x+3 & \text{ if } x<20, \\ 2x-2 & \text{ if } x\geq 20 . \end{cases} What is the value of f 1 ( 7 ) + f 1 ( 46 ) ? f^{-1}(7)+f^{-1}(46)?

27 25 26 28

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1 solution

Tom Engelsman
Nov 8, 2020

The required inverse function is just:

f 1 ( x ) = x 3 ( x < 20 ) , 1 2 x + 1 ( x 20 ) f^{-1}(x) = x-3 (x<20), \frac{1}{2}x+1 (x \ge 20)

Thus, f 1 ( 7 ) + f 1 ( 46 ) = ( 7 3 ) + ( 46 2 + 1 ) = 4 + 24 = 28 . f^{-1}(7) + f^{-1}(46) = (7-3) + (\frac{46}{2}+1) = 4 + 24 = \boxed{28}.

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