Haskell has a takeWhile function:
Prelude> takeWhile odd [1,3,5,7,8,9]
[1,3,5,7]
It “takes” elements from a list as long as applying the predicate function results in True. At the point it becomes False, it stops.
How can we implement it?
Here is a Haskell recursive approach I came up with:
takewhile::(a->Bool)->[a]->[a]
takewhile _ [] = []
takewhile f (x:xs) | f x == True = x : takewhile f xs
                   | otherwise = []
It keeps on calling itself as long as predicate f x is True, otherwise it returns an empty list without calling itself.
I could come up with the following implementation for JavaScript. It is a bit verbose and invokes defining another function to pass the intermediate result around:
function takeWhile(f, xs) {
 return take(f, xs, [])
}
function take(f, xs, arr) {
 if(!xs || xs.length === 0) {
 return arr
 }
 x = xs.shift()
 if(f(x)) {
   arr.push(x)
   return take(f, xs, arr)
 } else {
   return arr
 }
}
takeWhile((x)=>{
 return x % 2 !== 0
},[1,3,5,7,9,11])
Are there better ideas for implementing it in JavaScript?
If you want your takeWhile to perform like in HS, i.e. lazily, you need generators in JS:
function* takeWhile(fn, xs) {
    for (let x of xs)
        if (fn(x))
            yield x;
        else
            break;
}
function* naturalNumbers() {
    let n = 0;
    while (true)
        yield n++;
}
result = takeWhile(x => x < 10, naturalNumbers())
console.log([...result])
A straight port of the HS code is also possible, but it only works with materialized arrays (that is, eagerly):
// would be nice, but JS sucks ;(
// let takeWhile = (f, [x, ...xs]) => f(x) ? [x, ...takeWhile(f, xs)] : [];
let takeWhile = (f, xs) => xs.length ? takeWhileNotEmpty(f, xs) : [];
let takeWhileNotEmpty = (f, [x, ...xs]) =>  f(x) ? [x, ...takeWhile(f, xs)] : [];
let odd = x => x % 2
a = [1,3,5,7,8,9]
r = takeWhile(odd, a)
console.log(r)
Actually, as @naomik shows here there's a better way to deal with empty lists:
let nil = {};
let takeWhile = (f, [x = nil, ...xs]) => (x === nil || !f(x)) 
    ? [] : [x, ...takeWhile(f, xs)];
console.log(takeWhile(x => x % 2, [1, 3, 5, 7, 8, 9]));
Finally, your initial attempt does have a point, because, unlike the above, it's tail-recursive, which is a Good Thing. It can be written more concisely as
let takeWhile = (f, xs) => take1(f, xs, []);
let take1 = (f, xs, acc) => xs.length ? take2(f, xs, acc) : acc;
let take2 = (f, [x, ...xs], acc) => f(x) ? take1(f, xs, acc.concat(x)) : acc;
A combination of both approaches (that is, a recursive generator) left as an exercise...
Here's a concise solution:
const takeWhile = (fn, arr) => {
  const [x, ...xs] = arr;
  if (arr.length > 0 && fn(x)) {
    return [x, ...takeWhile(fn, xs)]
  } else {
    return [];
  }
};
Here's an extra condensed version of the above:
const takeWhile = 
  (fn, a) => a.length && fn(a[0]) ? [a[0], ...takeWhile(fn, a.slice(1))] : [];
It's not easily readable, so I don't recommend you use the condensed version.
A tail recursive version:
const takeWhile = (fn, arr) => {
  const recur = (a, acc) => {
    const [x, ...xs] = a;
    if (a.length > 0 && fn(x)) {
      return recur(xs, [...acc, x]);
    } else {
      return acc;
    }
  }
  return recur(arr, []);
};
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