For the function 37

Question

For the function ², find ^-1/

^-1  

Answer 1

Ok so first we have to find the inverse function f^-1(y), we do this by solving for x in terms of y. However, x is in two places we need to be a little bit clever to make sure that there is only one x on the right side of the equation. This calls for completing the square:

x²-4x = x²-4x+4-4 = (x-2)²-4

now x is in only one place so we can solve for the inverse with algebra

y = (x-2)²-4

y+4 = (x-2)²

√(y+4) = x-2

we only take the positive root on the left side because we set x>= 2 so the right side of the equation is positive so therefore the left side must be as well

x =√(y+4)+2

now we have our inverse function and can take the derivative with respect to y

we just use the power rule and find

(f^-1)^' = 1/2*(y+4)^-1/2 and (f^-1(4))^' = 1/2*8^-1/2= 1/(4*√2)

Of course check all the math yourself so you make sure you understand it and I did nothing wrong