Given f(x)=7-2(x-1)^2, x>= 1, determine f inverse (5).
To find the inverse function of f(x), we can follow these steps:
Step 1: Replace f(x) with y:
y = 7 - 2(x - 1)^2
Step 2: Swap x and y:
x = 7 - 2(y - 1)^2
Step 3: Solve for y:
x - 7 = -2(y - 1)^2
(x - 7)/-2 = (y - 1)^2
√[(x - 7)/-2] = y - 1
√[(x - 7)/-2] + 1 = y
Step 4: Replace y with f^(-1)(x):
f^(-1)(x) = √[(x - 7)/-2] + 1
Now, let's find f^(-1)(5) by substituting 5 into the inverse function:
f^(-1)(5) = √[(5 - 7)/-2] + 1
= √[-2/-2] + 1
= √1 + 1
= 1 + 1
= 2
Therefore, f^(-1)(5) = 2.