Question #8ec14

1 Answer
Nov 21, 2017

(x-1)/-x

Explanation:

(x-1)/(f(x) - f(x+1))

To solve this, you first have to simplify the denominator. In order to do this, you need to know what f(x) is.

They tell us that f(x+1) = x^2 +x. This is a start.

f(x) seems to equal x * k, where k means "some number that we don't know yet". To find k, we can rewrite the provided function as this: x^2 + x = (x + 1)k.
Now we see that multiplying (x + 1) by x will give us x^2 + x, so k = x.
(See that (x^2 + x)/x = x + 1.)

Now we can rewrite f(x) as f(n) = nx.
(I'm using n here where I would normally use x because it will keep us from getting confused; if I continued to use x, my function would say that f(x) = x^2, which is not true.)

Now we can plug this into the statement that we need to solve.
(x-1)/(f(x) - f(x+1)) = (x-1)/((x)(x) - (x+1)(x)
(x-1)/(x^2 - (x^2 + x))
(x-1)/(x^2 - x^2 - x)

x^2 - x^2 = 0, so we're left with

(x-1)/-x