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Let 

f(x)  = k(x) if x > 3

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

 

Find k(x) such that f is it's own inverse.

 Feb 7, 2021
 #1
avatar+112513 
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Let 

f(x)  = k(x) if x > 3

f(x) = (x - 3)^2 + 3 if x <= 3                                 x<=3,  y>3

 

Find k(x) such that f is it's own inverse.

 

y=(x - 3)^2 + 3

make x the subject

\(x=\pm\sqrt{y-3}+3\\ \text{For the inverse, swap x and y}\\ y=\pm\sqrt{x-3}+3 \qquad \\but \quad y\le3 \quad and \quad x>3\\ \text{ Since y is negative it must be the - sign}\\ y=-\sqrt{x-3}+3 \qquad where \qquad x>3 \)

 

so 

\(k(x)=-\sqrt{x-3}+3 \qquad where \qquad x>3\)

 

 Feb 7, 2021

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