Question

Determine if the two functions f and g are inverses of each other algebraically. If not, why not?

Answer 1

The given functions are

$f(x) = \frac{3}{x-5} , g(x) = \frac{3+5x}{x}$

First we find fog(x)= f(g(x))

So we need to substitute the value of g(x) for x in f(x), that is

$f(g(x)) = \frac{3}{\frac{3+5x}{x}-5} = \frac{3x}{3+5x-5x}$

$= \frac{3x}{3} = x$

Now we need to check the value of gof(x)

gof(x) = g(f(x))

So we need to substitute the value of f(x) in g(x), that is

$g(f(x))=\frac{3+5*\frac{3}{x-5}}{\frac{3}{x-5}}= \frac{3x-15+15}{3}$

$=\frac{3x}{3} = x$

And since

$fog(x) =gof(x) =x$

So the functions are inverse of each other .

Correct option is C .