To find the inverse of the function, simply switch x and y and solve for y.
Starting from the original function, f(x) = (x - 4) / (x + 2), we swap x and y to get x = (y - 4) / (y + 2).
From there, we solve for y:
1. Multiply each side by (y + 2) to get rid of the denominator: x * (y + 2) = y - 4.
2. Distribute x to y and 2 on the left side: xy + 2x = y - 4.
3. Rearrange the equation to group terms containing y together: xy - y = -2x - 4.
4. Factor out y on the left side: y * (x - 1) = -2x - 4.
5. Dividing each side by (x - 1) to isolate y gives the inverse function: y = -2(x + 2) / (x - 1) or f^(-1)(x) = -2(x + 2) / (x - 1).
Now, let's find the domain and range of each function.
The domain of a function is all possible values of x for which the function is defined. For the original function, f, it's undefined when the denominator equals zero. Therefore, x ≠ -2. In interval notation, we write this as (-∞, -2) ∪ (-2, ∞).
Likewise, the range of a function is the set of all possible output values (y-values), which is derived from the domain of the function. Given the nature of the original function, its range is all real numbers except 2. Thus, the range of f is (-∞, 2) ∪ (2, ∞).
Since an inverse function switches the roles of x and y, the domain of the inverse function is the range of the original function, and its range is the domain of the original function.
Therefore, the domain of the inverse function f^(-1) is (-∞, 2) ∪ (2, ∞) and the range of f^(-1) is (-∞, -2) ∪ (-2, ∞).