Answer: a. We can factor the quadratic function f (x) = 8 x^2 - 16 as follows: f (x) = 8 x^2 - 16 = 8 (x^2 - 2) = 8 (x - √2)(x + √2).
b. The x-intercepts of f (x) occur when x = 0, so the x-intercepts are at x = √2 and x = -√2.
c. To find f (-1), we simply plug in -1 for x in the equation f (x) = 8 (x - √2)(x + √2) to get f (-1) = 8 (-1 - √2)(-1 + √2) = 8 (-1 - √2)(-1 - √2) = 8 (-2 - 2√2) = -64 - 64√2.
d. The domain of f (x) is the set of all real numbers, since the quadratic equation f (x) = 8 (x - √2)(x + √2) is defined for all real values of x. The range of f (x) is also the set of all real numbers, since the quadratic equation can produce any real value for f (x) given a real value of x.
e. The inverse of f (x) is denoted as f^(-1)(x), and it is defined as the function that "undoes" the original function. In other words, if we apply the inverse function to the result of the original function, we should get back the original input. For example, if we apply the original function f (x) to some value x, and get the result y, then applying the inverse function to y should give us x back. In this case, the inverse of f (x) is the function f^(-1)(x) = (1/8) x + √2.
f. The domain of the inverse function f^(-1)(x) is the range of the original function f (x), and the range of the inverse function is the domain of the original function. In this case, the domain of f^(-1)(x) is the set of all real numbers (since the range of f (x) is the set of all real numbers), and the range of f^(-1)(x) is also the set of all real numbers (since the domain of f (x) is the set of all real numbers).
g. The relationship between the domain and range of a function and its inverse is that the domain of the inverse is the range of the original function, and the range of the inverse is the domain of the original function.
h. Not all functions and their inverses will have the same relationship between their domains and ranges. For example, a function that is defined only for a certain range of values (such as a square root function, which is only defined for non-negative values of x) will have a restricted domain, and its inverse will have a corresponding restricted range.