SOLUTION: Let f(x) = (x-2)^3+8 a. Show that this function is one-to-one algebraically. b. Find the inverse of f(x).

Question 1096389: Let f(x) = (x-2)^3+8
a. Show that this function is one-to-one algebraically.
b. Find the inverse of f(x).
Answer by htmentor(1343) (Show Source): You can put this solution on YOUR website!
a. The function is one-to-one if we let f(a) = f(b) and are able to show that a = b.
f(a) = (a-2)^3 + 8
f(b) = (b-2)^2 + 8
f(a) = f(b): (a-2)^3 + 8 = (b-2)^2 + 8
(a-2)^3 = (b-2)^3
Let A = a-2, B = b-2
If A^3 = B^3 -> A = B, and therefore a = b
b. To find the inverse of f(x) = y = (x-2)^3+8, replace x with y and solve for y:
x = (y-2)^3 + 8
y-2 = (x-8)^(1/3)
y = (x-8)^(1/3) + 2
The inverse is f_inv(x) = (x-8)^(1/3) + 2

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