The function f(x)=7/12 x^2+¦Ð is one-to-one on the domain if and if x¡Ý0. Find the f^(-1) (x).
To find the inverse of the function f(x), denoted as f^(-1)(x), we can follow these steps:
Step 1: Replace f(x) with y
y = (7/12)x^2 + ¦Ð
Step 2: Swap the roles of x and y
x = (7/12)y^2 + ¦Ð
Step 3: Solve the equation for y
Subtract ¦Ð from both sides:
x - ¦Ð = (7/12)y^2
Step 4: Multiply both sides by 12/7 to isolate y^2
(12/7)(x - ¦Ð) = y^2
Step 5: Take the square root of both sides
±√((12/7)(x - ¦Ð)) = y
Step 6: Simplify the expression by taking ±√ separately
y = ±√((12/7)(x - ¦Ð))
So, the inverse function f^(-1)(x) is:
f^(-1)(x) = ±√((12/7)(x - ¦Ð))
Please note that the ± symbol indicates that for any given x in the domain, there are two possible values for f^(-1)(x).