Answer:
(a) To find the range of the function f(x), we need to consider all possible values of x in the range -7 ≤ x ≤ 7 and x ≠ -5. For each value of x, we can evaluate the function to find the corresponding value of f(x). Then, we can find the minimum and maximum values of f(x) in this range to determine the range of the function.
For example, if x = -7, then the function will return a value of 2 - 12/(-7) + 5 = 2 + 2 + 5 = 9. If x = 7, then the function will return a value of 2 - 12/7 + 5 = 2 - 1 + 5 = 6.
We can see that the minimum value of the function is 6, which is achieved when x = 7, and the maximum value of the function is 9, which is achieved when x = -7. Therefore, the range of the function is {6, 9}.
(b) To find the inverse of the function f(x), we need to switch the x and y values in the function definition to obtain a new function g(y) such that g(f(x)) = x for all values of x in the domain of the original function.
In this case, the inverse of the function f(x) is given by g(y) = (2 - y + 5)/(-12). This inverse function is defined for all values of y in the range {6, 9}, which is the range of the original function.
To find the value of f^-1(0), we can plug in 0 for y in the inverse function g(y) to obtain g(0) = (2 - 0 + 5)/(-12) = -1/12. Therefore, f^-1(0) = -1/12.