Final answer:
To rewrite the rational function using long division, divide the numerator by the denominator. The fully simplified form of the function is f(x) = 7 + (-21 / (x + 3)).
Step-by-step explanation:
To rewrite the rational function using long division, we can divide the numerator by the denominator.
First, write the function in the long division format:
7x x + 3
The first step is to divide 7x by x, which gives us 7. Write this above the line. Then, multiply 7 by x + 3 to get 7x + 21. Subtract this from 7x to get -21. Bring down the next term, which is 0. Repeat the process until there are no more terms to bring down.
7 x + 3
Since the remainder is -21 and there are no more terms to bring down, we can write the rational function as:
f(x) = 7 + (-21 / (x + 3))
This is the fully simplified form of the rational function using long division.