f(x) = (6x+12)^(1/2).
To find the inverse function of f(x).
Let f^(-1) (x) = y be the inverse function of f(x).
The x = f(y).
By definition, f(y) = put y in place of x iin (6x+12)^1/2).
=> f(y) = (6y+12)^(1/2).
=> x = (6y+12)^(1/2).
We square both sides:
x^2 = 6y+12.
x^2-12 = 6y.
Therefore y = 1/6(x^2-12).
Therefore f^(-1) (x) = y = (1/6)x^2 - 2 is the inverse of f(x). f^(1) (x) = y i= not 1/sqrt(6x+12).
f^-1(x) is the inverse function of f(x).
First, we'll note y= sqrt(6x+12).
Now we'll change x by y:
We'll raise to square both sides to eliminate the square root:
x^2 = 6y + 12
We'll isolate y to the left side:
-6y = 12 - x^2
We'll divide by -6:
y = x^2/6 - 2
The inverse function is:
f^-1(x) = x^2/6 - 2
We notice that the expression of the inverse function is not the inversed of the expression of the original function!