Question

Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x.

f(x) = x^3 + 4 and g(x) = cube root of (x-4)

Answer 1

So, first try f(g(x)), this means put the result of function g in function f:\[f(g(x))=\left( \sqrt[3]{x-4} \right)^3+4=...\]Can you see what the next step is?

Answer 2

no. :(

Answer 3

What is the definition of the 3rd root of a number?

Answer 4

The number whose cube is equal to a given number

Answer 5

OK, but not only a given number, but the number from which you wanted to know the 3rd root: e.g: the 3rd root of 8 is 2, because: 2^3=8.

So: the 3rd root of x-4 is p, then p³=x-4. So...\[(\sqrt[3]{x-4})^3=x-4\]by definition.

Answer 6

so that will get the expression out from under the radical

Answer 7

so i'll get x-4+4

Answer 8

or x-8

Answer 9

@ZeHanz am i correct?

Answer 10

You get x-4+4=x in my book...

Answer 11

oh srry

Answer 12

So we've done the first half of the proof.
Now the second: put the result of f in g:\[g(f(x))=\sqrt[3]{x^3+4-4}\]

Answer 13

ok. thnx. but where did u get p from?

Answer 14

p is not important. You can name it anyway you want.
Just look at it this way: say you've got a terribly complex expression, such as:\[3x^{23}-5x^{16}+\frac{ 1 }{ 2x }\]
First, they want you to calculate the 3rd root of it.
I really don't know what it would be, so let's call it p for now:\[p=\sqrt[3]{3x^{23}-5x^{16}+\frac{ 1 }{ 2x }}\]Looks awful, not?

Now you have to calculate .... p³. Well, you don't have to think long about it, youve got your original complex expression back!

The reason for this, that the 3rd root and the 3rd power are each other's inverse.
This means: they undo each other's effect on numbers you put in!