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.
+118690 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
\(k(x)=-\sqrt{x-3}+3 \qquad where \qquad x>3\)
