The function replaces lowercase 'a', uppercase 'A', and lowercase 'e' with 'd' in the input string. All other characters remain unchanged.

[CODE]: 
```python
def approx_function(x):
    output = ''
    for char in x:
        if char in ['a', 'A', 'e']:
            output += 'd'
        else:
            output += char
    return output
```

Please let me know if you would like me to test more input cases to refine the description and approximation.