Y= (4/x-4) -3 find the domain and range?

To find the domain and range of the function Y = (4/x-4) - 3, we need to consider the restrictions on the variables x and y.

Domain:
The function is defined for all real numbers except when the denominator of the fraction is equal to zero. In this case, x-4 cannot be equal to zero, so x cannot be equal to 4. Therefore, the domain of the function is all real numbers except x = 4.

Range:
To find the range of the function, we need to consider the behavior of the function as x approaches positive and negative infinity. As x approaches positive infinity, the value of the function approaches -3. As x approaches negative infinity, the value of the function also approaches -3. Therefore, the range of the function is all real numbers except y = -3.

In summary:
Domain: All real numbers except x = 4
Range: All real numbers except y = -3

Domain:

The domain of a function is the set of all possible values of the independent variable for which the function is defined.

In this case, the independent variable is x. The function is defined for all values of x except for x = 4, since division by zero is undefined.

Therefore, the domain of the function is:

```
Domain: All real numbers except x = 4
or
Domain: (-∞, 4) ∪ (4, ∞)
```

Range:

The range of a function is the set of all possible values of the dependent variable.

To find the range of the function, we can rewrite it in the form:

```
y = 4/(x-4) - 3
y = 4/(x-4) - 3(x-4) / (x-4)
y = 4/(x-4) - (3x-12) / (x-4)
y = (4-3x+12) / (x-4)
y = (16-3x) / (x-4)
```

From this form, we can see that the function is undefined for x = 4. However, we can also see that the function is continuous for all other values of x.

Therefore, the range of the function is:

```
Range: All real numbers except y = 16/3
or
Range: (-∞, 16/3) ∪ (16/3, ∞)
```