Find a formula for the inverse of the following function, if possible.

Question

KyledaGreat · Answer

$$S(x) = (x^3 + 5)^{\frac{ 1 }{ 5}} - 1$$

$$s^{-1}(x) =$$

Tranquility · Answer

switch the x and y's and what do you get?

KyledaGreat · Answer

y = (x^3 + 5)^1/5 - 1

Tranquility · Answer

Now switch the x and y

KyledaGreat · Answer

x = (y^3 + 5)^1/5 - 1

Tranquility · Answer

You properly changed the S(x) to y but now you need to switch it

Tranquility · Answer

Now add 1 to both sides

KyledaGreat · Answer

that's what i did

Tranquility · Answer

Yes, sorry. I had sent that right as you sent it

KyledaGreat · Answer

it's okay

KyledaGreat · Answer

the answer didn't come in

Tranquility · Answer

x = (y^3 + 5)^1/5 - 1

We're still trying to get y on one side of the equal sign and everything else on the other side

What do you get when you add 1 to both sides?

KyledaGreat · Answer

$$y = \sqrt[3]{x^5 + 5x^4 + 10x^3 + 10x^2  + 5x - 4}$$

Tranquility · Answer

you're probably plugging it all into a calculator, aren't you?

Tranquility · Answer

which is why your answers are expanded all the way

KyledaGreat · Answer

yes actually on mathway

KyledaGreat · Answer

Add 1 both sides

$$(y^3 + 5)^{\frac{ 1 }{ 5 }} - 1 + 1 = x + 1$$

Tranquility · Answer

and then the -1 + 1 cancel out on the left side and you're left with

$ x + 1 = (y^3 + 5)^{\frac{1}{5}} $

Now you raise everything to the power of 5 on both sides to cancel out the 1/5 and you get

(x+1)^5 = y^3 + 5

Tranquility · Answer

Next, what do you get when you subtract 5 from both sides?

KyledaGreat · Answer

(x+1)^5 = y^3 + 5 - 5 =

like this ?

KyledaGreat · Answer

It is $$x = \sqrt[5]{y^3} - 1$$

Tranquility · Answer

@kyledagreat wrote:

(x+1)^5 = y^3 + 5 - 5 = like this ?

You forgot to subtract 5 on the other side as well so (x+1)^5 - 5= y^3 + 5 - 5 Whatever you do on one side, you have to do on the other side so that way it remains equal and you're not changing the equation. You're only re-arranging the terms. The +5 - 5 on the right side will become 0 and so what are you left with?

KyledaGreat · Answer

Simplify

$$(x + 1)^5 = y^3 + 5$$

KyledaGreat · Answer

Solution: $$x = \sqrt[5]{y^3 - 1}$$

KyledaGreat · Answer

i see what you mean , i thought on the end row to do both sides

Tranquility · Answer

I don't know where you're getting all that from

Tranquility · Answer

@kyledagreat wrote:

Simplify $$(x + 1)^5 = y^3 + 5$$

We need to solve for y and we need to do it one step at a time. If you use a calculator, it gives you an expanded answer which isn't correct Subtract 5 on both sides and you get (x+1)^5 - 5 = y^3 The last step is to take the cube root of both sides so that you'll have y = something

KyledaGreat · Answer

ok you mean to do it like this:

KyledaGreat · Answer

ok you mean to do it like this: $$y = (x+1)^5 - 5 = y^3$$

KyledaGreat · Answer

but wait i can't enter the = sign

Tranquility · Answer

You can't have two equal signs

Tranquility · Answer

You also forgot to take the cube root of all that on the right side

Tranquility · Answer

$y = \sqrt[3]{(x+1)^5 - 5 }$

KyledaGreat · Answer

attachment

KyledaGreat · Answer

is that right

Tranquility · Answer

should be

KyledaGreat · Answer

that's right, thank you so much for all of your help Tran for today man. I appreciate it so much

Tranquility · Answer

I'm glad I could help!

1 attachment