Suppose that the functions f and g are defined as follows Fx

Suppose that the functions f and g are defined as follows. F(x) = 9/x + 4 g(x) = 5/x Find f/g. Then, give its domain using an interval or Union of intervals. Simplify your answers. (f/g) (x) = Domain of f/g:

Solution

f(x) = 9/(x + 4) ; g(x) = 5/x

(f/g)(x) = f(x) / g(x)

==> (f/g)(x) = [9/(x + 4)] / [5/x]

==> (f/g)(x) = 9x/(5(x+4)) ; since (a/b)/(c/d) = ad/(bc)

Domain ==> the values of x for which the function is defined.

(f/g) (x) = 9x/(5(x+4))

It is undefined if denominator is zero

==> 5(x + 4) = 0

==> x = -4

and also as the function g(x) is in denominator of (f/g)(x), (f/g)(x) is undefined if g(x) is undefined.

g(x) is undefined for x = 0 since the denominator becomes zero and the function is undefined if the value of denominator is zero.

==> (f/g) (x) is undefined for x = -4 and x = 0

Hence domian of the function (f/g)(x) is (-, -4) U (-4 , 0) U (0 , )