Final answer:
The function f(x) = 1/(3x+7) is composed of a translation by 7 units, a horizontal scaling of 3, and an inversion. The inverse of this function is found by solving y = 1/(3x+7) for x, resulting in f^-1(y) = (1/y - 7)/3. A negative exponent represents an inversion, such as x^-1 = 1/x.
Step-by-step explanation:
To express the function f(x) = 1/(3x+7) as a composition of translations, scalings, and inversions, we can sequentially apply transformations to the simplest function, which is f(x) = 1/x. The first step is to apply a horizontal scaling of 3 to the argument x, resulting in f(x) = 1/(3x). Next, we translate it horizontally by 7 units, obtaining f(x) = 1/(3x+7).
The inverse of a function essentially undoes what the original function does. For f(x) = 1/(3x+7), to find its inverse, we set f(x) equal to y and solve for x in terms of y. This process gives us the inverse function f-1(y) = (1/y - 7)/3.
Understanding that a negative exponent is equivalent to an inversion is critical. For instance, x-1 = 1/x is an example of an inversion, which we applied to our original function.