Solve Inverse Function of g(x)=(x2/e)+2 ln(x)-e

[raptik]

Homework Statement

The function g(x) = (x²/e) + 2 ln(x) - e on (0,infinity) is one-to-one. Evaluate g^-1(2)

Homework Equations

Find x in terms of y. Then switch x and y. Plug in 2 to the new equation.

The Attempt at a Solution

I can think of no way to get x explicitly in terms of y. I considered plugging in 2 to the original equation to get (x₁,y₁) and switching the two, but without a calculator, it seems unlikely to find a straightforward answer. Please help.

 

[╔(σ_σ)╝]

The question only ask you to evaluate $g^{-1}(2)$ it did not ask for a formula.

Think about what an inverse is if f(x) =y then $f^{-1} (y) = ?$.

Inverse "brings things back". If I throw a ball at you and "inverse" is you throwing the same ball at me, right ?

 

[raptik]

The answer choices are: A) 1 B) 2 C) e D) e² E) 0

One of these is the correct "evaluation".

I follow that if f(x) =y then f^-1 (y) = x

But when I don't know how to get to x, I'm not going to be able to solve this.

 

[╔(σ_σ)╝]

Have you considered finding g(e) ?

 

[raptik]

Have you considered finding g(e) ?

Oh!

If I put x=e, then I get g(e) = 2. So (e,2). Then it's inverse is (2,e) which matches with g^-1(2) to give me e. I see how that could work, but how did you have the intuition to add e to the original problem? I suppose it's a matter of plugging something that seems like it would give me the required value until it works. Thnx for help.

 

[╔(σ_σ)╝]

Oh!

If I put x=e, then I get g(e) = 2. So (e,2). Then it's inverse is (2,e) which matches with g^-1(2) to give me e. I see how that could work, but how did you have the intuition to add e to the original problem? I suppose it's a matter of plugging something that seems like it would give me the required value until it works. Thnx for help.

When i looked at g i knew that the x^2 term would cause problems. I looked again at g and realized there was a 2 infront of lnx and i knew ln(e) =1. From there i realized that if i found g(e) i would get 2. ;-)