Question

f(x)=1/x-2 , g(x)=2x+1/x

A. Use composition to prove whether or not the functions are inverses of each other.
B. Express the domain of the compositions using interval notation.

Answer 1

Answer:

A. They are inverses of each other.

Step-by-step explanation:

A.

If they are inverses of each other then f(g(x)) will be = x.

f(g(x)) =   1 /  ( (2x + 1)/ x) - 2)

= 1 / ( 2x + 1 - 2x)/ x

= 1 /  1/x

= x.

So they ARE inverses of each other.

Answer 2

Answer:

Yes, they are inverse of each other.

Domain = $(-\infty, \infty)$

Step-by-step explanation:

We  re given the following:

$f(x) = \frac{1}{x-2}\\g(x) = \frac{2x+1}{x}$

If we calculate the composite function, it will be of the form:

$f(g(x)) = \frac{1}{\frac{2x+1}{x}-2} \\= \frac{x}{2x + 1 - 2x}\\= x$

$g(f(x)) = \frac{2(\frac{1}{x-2}) + 1}{x-2}\\= \frac{2+x-2}{1}\\\\= x$

Since, f(g(x)) = g(f((x)) = x, the functions are inverse of each other.

The domain of composite functions, f(g(x)) and g(f((x)) are the values that x can take, so the domain for composite number is all real numbers that is $(-\infty, \infty)$