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Your example is a bit too trivial to know what you're asking. The naive answer is

function name(opt) {
  return opt;
}

Are you looking for the ternary operator?

function name(opt) {
  return (opt === 'cat')
    ? 'cat'
    : (opt === 'dog')
      ? 'dog'
      : undefined;
}

Or something like a string enum?

const Animals = {
  cat: 'cat',
  dog: 'dog'
};

Animals.cat; // 'cat'
Animals.dog; // 'dog'

Or a real functional programming concept like a curried function? (This is not the best example though)

function name(opt) {
  return opt || (arg) => name(arg);
}

name('cat') // 'cat'
name()('dog') // 'dog'