Determining the inverse function for a given function

[steve snash]

Homework Statement

Determine the inverse function for the function,

f(x)=((2x+1)^3)-2

I think i know the steps but i want to know if my answer is correct
1.Change f(x) to f(y)
2.Re write so x=f(y)
3.Swap x and y variables
4.replace y with f^-1(x)

The Attempt at a Solution

1.y=((2x+1)^3)-2
2.cube sqrt y=(2x+1)-2
=cube sqrt y+2=2x+1
=cube sqrt y+1=2x
=(cube sqrt y+1)/2=x
3.y=(cube sqrt x+1)/2
4.f^-1(x)=(cube sqrt x+1)/2

Is this right? Can someone please tell me how to do it if its wrong?

 

[Mentallic]

It would be easiest if you just switch the x and y variables at the start (don't deal with f(x) and f(y) etc. while solving the equation), and then once you found the answer, change your y into f^-1(x).

y=((2x+1)^3)-2
cube sqrt y=(2x+1)-2

No! $$\sqrt[3]{((2x+1)^3-2}\neq (2x+1)-2$$

To illustrate this, I'll give you a more basic example that's a common mistake. Take pythagoras' theorem: $$c=\sqrt{a^2+b^2}$$. Now, many students that come across this formula outside of solving lengths for right-triangles often make the mistake that $$\sqrt{a^2+b^2}=a+b$$ since they can just take the square root of each term inside... WRONG!
Obviously if this were true we would have simplified pythagoras' theorem to c=a+b

The rule is $$\sqrt{x^2}=x$$ and x can be just about anything you choose, such as a+b. But then $$(a+b)^2=a^2+2ab+b^2\neq a^2+b^2$$. Only until you can convert $$(2x+1)^3-2$$ into $$a^3$$ where a is some variable involving x, then can you take the cube root.

But this is impossible, so all you need to do is move the -2 over to the other side, then you can take the cube root.

 

[steve snash]

So how do you change the (2x+1)^3-2 so x=y (eg) how do you find the answer?? I am confused :p

 

[muso07]

I remember also thinking that inverse functions were tricky when I started.

Your function is y = (2x+1)³-2.

To find the inverse, switch x and y so that you get x = (2y+1)³-2.

Now the tricky bit is rearraging this so you have it in terms of y. So we need y = ...

First add 2 to both sides, then you'll be on your way.

 

[HallsofIvy]

Homework Statement

Determine the inverse function for the function,

f(x)=((2x+1)^3)-2

I think i know the steps but i want to know if my answer is correct
1.Change f(x) to f(y)
2.Re write so x=f(y)
3.Swap x and y variables
4.replace y with f^-1(x)

The Attempt at a Solution

1.y=((2x+1)^3)-2
2.cube sqrt y=(2x+1)-2

You have the order wrong. If you were evaluating this for some specific x, you would subtract 2 last so finding the inverse, you add 2 first:
$y+ 2= (2x+ 1)^3$. Now take the cube root of both sides.

=cube sqrt y+2=2x+1

And please don't say "cube sqrt"- the "sqrt" or "square root" is specifically the inverse of squaring. What you want here, the inverse of cubing is "cube root".

=cube sqrt y+1=2x

You should have, at this point, cube root(y+ 2)= 2x+1. Solve that for x.

=(cube sqrt y+1)/2=x
3.y=(cube sqrt x+1)/2
4.f^-1(x)=(cube sqrt x+1)/2

Is this right? Can someone please tell me how to do it if its wrong?