How to find inverse of y = x^2 + x^1/2

[dwsingrs]

Greetings.

Question: how to calculate the inverse of y = x² + x^1/2. It's not a quadratic, like y = x^1/2 + x^1/4, where one can complete the square.

I've been obsessing all week over this, but don't give me the answer on a silver platter (yet) - just a hint.

So far:

y = x² + x^1/2

y - x² = x^1/2

(y - x²)² = (x^1/2)² = x

Therefore:

y - x² = x^1/2 and y - x² = - (x^1/2)

(-1) (y - x²) = (-1) (x^1/2) (Any reason why I can't pick one or the other of these equations and mult. both sides by -1, maintaining the equality and the expression?)

(Plus, for what it's worth, y = x² + x^1/2 and y = x² - x^1/2 give the same y value only when x = 0. It seems all I'm showing is where the two graphs intersect.)

(Anyway)

Therefore:

x² - y = -(x^1/2)

Therefore:

x² - y = y - x²

2x² = 2y

x² = y

x = y^1/2 and x = -(y^1/2)

ASSUMING that this is correct, if I want to find the inverse of the inverse (i.e., go back to y = f(x), y = x² + x^1/2), I don't see how to get there starting with either x = -(y^1/2) or x = y^1/2. You get as far as y = x², and think you're done, IF you have no prior knowledge of y = x² + x^1/2.

What am I missing?

 

[tkhunny]

You can go through the algebra for practice. That might be a great idea.

You could just go straight to the quadratic formula.

\(\displaystyle x^{\frac{1}{2}} + x^{\frac{1}{4}} - y = 0\)

So that:

\(\displaystyle x^{\frac{1}{4}} = \frac{-1\pm\sqrt{1-4(1)(-4)}}{2}\)

Looks to me like maybe your completing the square wandered off, somewhere.

 

[soroban]

Hello, dwsingrs!

I don't think it can be done . . .

Find the inverse of: .\(\displaystyle y \:=\: x^2 + x^{\frac{1}{2}}\)

Switch variables: . \(\displaystyle x \:=\:y^2 + y^{\frac{1}{2}}\)

. . . . . . . . . .\(\displaystyle x - y^2 \:=\:y^{\frac{1}{2}}\)

Square: . .\(\displaystyle (x-y^2)^2 \:=\:\left(y^{\frac{1}{2}}\right)^2\)

. . . \(\displaystyle x^2 - 2xy^2 + y^4 \:=\:y\)

\(\displaystyle y^4 - 2xy^2 - y + x^2 \:=\:0\)


Now solve for \(\displaystyle y.\)

. . Good luck!

 

[tkhunny]

Wow. Partial reading strikes again.