First of all, check [here](http://support.wolfram.com/kb/3820) and [here](http://mathematica.stackexchange.com/questions/18393/what-are-the-most-common-pitfalls-awaiting-new-users/26037#26037).  This is a common pitfall, and questions related to this are posted literally weekly, so I am going to let you review those articles.

In short, if you try evaluating `medianLEfromQx[x qx1]` with `x` having no value, you'll see that it returns a number.  This expression evaluates inside `FindRoot` even before `FindRoot` gets a chance to substitute a value for `x`.  So you would have to make `medianLEfromQx` not evaluate except for truly numerical vector arguments.

You can do this by changing its definition to look like:

    Clear[medianLEfromQx]
    medianLEfromQx[qx_ /; (VectorQ[qx, NumericQ])] := ...

Now `medianLEfromQx[x qx1]` won't evaluate unless `x` has a numerical value.

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Next, `Solve` and `NSolve` won't work on numerical blackboxes, only `FindRoot` will.  `Solve` only works with symbolic equations with exact coefficients. `NSolve` is designed for solving polynomial equations (or equations that can be reduced to a polynomial equation) numerically, thus it also needs to see the structure of an equation and won't work with a numerical black box.

So the only candidate here is `FindRoot`.

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However `FindRoot` isn't very appropriate here either. The methods it can use all assume that the function they're working with is a "nice and smooth one".  Your function always returns integers, so it has a "step structure".  The default `FindRoot` method would try to approximate the derivative of the function and would of course fail: the derivative is zero everywhere.

You *can* use [Brent's method][1], but this isn't ideal either:  `FindRoot[medianLEfromQx[x*qx1] == 8, {x, 0, 2}, Method -> "Brent"]`

Instead I would just plot the function and visually check the range of `x` which satisfies this equation.

    Plot[medianLEfromQx[x qx1] - 8, {x, 0, 10}]

![enter image description here][2]


  [1]: http://en.wikipedia.org/wiki/Brent%27s_method
  [2]: https://i.sstatic.net/pPsa9.png