f(1/x+1) = 2x-12, compute f(0)

Question

for all real numbers x

Answer 1

f(1/(x+1))=2x-12.

(i) Let y=1/(x+1), xy+y=1, x=(1-y)/y;

f(y)=2(1-y)/y-12=(2-2y-12y)/y=(2-14y/y=(2/y)-14, which can also be written:

f(x)=(2/x)-14.

so f(0)=2/0-14⇒∞.

(ii) Another way:

If 1/(x+1)=0, then x→∞, so f(0)→∞.

[To find f(2) we have:

(i) f(2)=(2/2)-14=-13

(ii) 1/(x+1)=2, 2x+2=1, x=-½, so f(2)=2(-½)-12=-13.

This shows that the two methods yield the same result.]