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Let

f(x) = k(x) if x > 2

f(x) = -3x + 8 if x <= 2

Find the function k(x) such that f(x) is its own inverse.

 Jan 2, 2022
 #1
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It really helps to graph on this question.  

 

Essentially, a function is its own inverse if it can be reflected over the line and still contain the exact same values.  

 

If we graph f(x) = -3x + 8 if x <= 2, we have something that passes through (2, 2), and (0, 8).

 

Thus, k(x) must be the reflection of the line -3x+8 over the line y=x, meaning that it must pass through (2, 2) and (8, 0)

 

The line that passes through these points is k(x)= -1/3x + 8/3.

 

If k(x) equals this, then the graph is symmetric about y=x, and thus, is its own inverse.  

 Jan 2, 2022

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