Let f(x)=(x-2)^3 - 8
a. Show that this function is one-to-one algebraically.
b. Find the inverse of f(x).
+129839 a. Show that this function is one-to-one algebraically
If the function is one-to-one, f(x) ≠ f(-x) and we have
(x - 2)^3 - 8 = ( (-x) - 2)^3 - 8 ???? simplify
(x - 2)^3 = ( [-1] (x + 2))^3 ????
(x - 2)^3 = (-1)^3 (x + 2)^3 ????
(x - 2)^3 = - (x + 2)^3 ????
x^3 - 6x^2 + 12 x - 8 = -x^3 - 6x^2 - 12x - 8 ???? and these are clearly unequal, so the function is one-to-one
b. Find the inverse of f(x)
For f(x), write y
y = (x - 2)^3 - 8 add 8 to both sides
y + 8 = (x - 2)^3 take the cube root of both sides
∛(y + 8) = x - 2 add 2 to both sides
∛(y + 8) + 2 = x "exchange" x and y
∛(x + 8) + 2 = y for y, write f-1(x)
∛(x + 8) + 2 = f-1(x)
