Question

Let f(x)=(x-2)^3 + 8

a. Show that this function is one-to-one algebraically.

b. Show the inverse of f(x).

Answer 1

a. If you're in calculus, you could find the derivative, and show the function is always increasing. That'd show it's one to one.

If you're not in calculus... idk if i have any ideas.

b. is easy. Just swap places with x and f(x), and solve for f(x).

Answer 2

Assuming you are in calculus, find the derivative, set equal to zero, solve for x. There'll be one solution.

Then you can plug values into the derivative to show it's always positive (or zero at that one value of x you just found) - check a value of x less than the value you found, and a value greater than it.

Answer 3

Let us assume that f(x) is equal for two different values x & y.

f(x)=f(y)
(x-2)^3+8=(y-2)^3+8
on solving it further you will find x=y.

Thus this contradicts our assumption that x & y are different number & f(x)=f(y).

Answer 4

if function is one to one if \(f(a) =f(b)\) implies \(a=b\)

So assume \((a-2)^3+8=(b-2)^3+8\) and show that that leads to \(a=b\)

Answer 5

I see now you wrote basically the same thing, but no need for a contradiction.

Answer 6

f(x)=y=(x-2)^3+8

convert it into form x=p(y).
where p(y) is another function