Pre-Calc Question

Question

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 1

Proof By contradiction

Suppose there are two DIFFERENT real values of x, x1 and x2, such that f(x1) = f(x2)

and x1 is NOT equal to x2.

Then

f(x1) = f(x2)

(x1 - 2)^3 + 8 = (x2 - 2)^3 + 8

(x1 - 2)^3 = (x2 - 2)^3 <--- subtracts the 8 from both sides

(x1-2) = (X2-2) <---- takes cube root of both sides

x1 = x2 <--- contradiction

INVERSE:

swaps X and y: x = (y-2)^3 + 8

x - 8 = (y-2)^3

cube_root(x-8) = y - 2

cube_root(x-8) + 2 = y