Inverse natural log question?

Question

I'm fine with basic inverse functions but I am still having trouble with some ie e^(2-sqrt(x)). Working backwards I end up with ln2+x^2 but I know thats incorrect. Please advise, thanks

Answer 1

y=e^(2-√x)
ln y = 2 - √x
√x = 2 - ln y
x = (2-ln y)²

So now just swap the x and y -- if f(x) = e^(2 - √x), then f⁻¹(x) = (2 - ln x)².

Edit: to answer the question you asked in e-mail -- the domain of the inverse function is not (0, ∞). The domain of the inverse function is only the range of the original function, which is (0, e²). If x > e², then 2 - ln x < 0, so √((2 - ln x)²) = ln x - 2, not 2 - ln x, and so f((2 - ln x)²^2) = e^(2 - sqrt ((2 - ln x)²)) = e^(2 - ln x + 2) = e^(4 - ln x) ≠ x.

This illustrates an important point -- despite what they teach you in those "find the domain of (formula)" exercises, the domain of a function is NOT determined by the formula that you plug numbers into, but rather is part of the definition of the function itself. Thus if a function is defined as the inverse of another function f, then its domain is the range of f, period -- even if the formula derived for the inverse might be applicable to numbers outside of f.

As a side note, I should also point out the fallaciousness of giving students a formula and asking, say, "what is the domain of 1/x?" 1/x is a formula, not a function. Asking for the domain of 1/x is rather like asking "who is the current president?" Strictly speaking, the correct answer is "the president of what? Microsoft? France?" Of course, someone asking that on a test would expect you to know that they meant "president of the united states", and in math they expect you to know that 1/x refers to the function defined on the largest subset of R for which 1/x is a valid formula, but it's still an ill-formed question, and very few teachers do a good job of explaining that, which leads to confusion when a function is defined to be on a smaller domain (say, the range of f), but the formula giving that function would make sense on a set larger than the domain of the function, as is the case here. Frankly, I wish that they would do away with those exercises altogether, but mathematical correctness has never been a high priority for those in charge of mathematics education.

Answer 2

5^2x + 5^x - 12 = 0 permit u=5^x u^2 + u - 12 = (u+4)(u-3) = 0 u = -4, u = 3 u = -4 = 5^x, take LN of the two facets, yet can not LN a adverse style. u = 3 = 5^x, take LN of the two facets, LN3 = LN5^x = xLN5, x = LN3/LN5 = .682606

Answer 3

Here is how you should have worked backwards, in my opinion:

e^(2-sqrt(x)) = y
2-sqrt(x) = ln y
2 - ln y = sqrt(x)
(2 - ln y)^2 = x