The function takes an input string and performs a substitution on each character using a specific mapping consistent across both upper and lower case letters. Digits and special characters remain unchanged.

[CODE]:
```python
def approximate_function(input_string):
    mapping = str.maketrans('abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ',
                            'myrugcztdfplojsbevnkaihwqxMYRUGCZTDFPLOJSBEVNKAIHWQX')
    return input_string.translate(mapping)
```