Let f x x 8 and gx square root of x1 a Find fg5 Show work b

Let f (x) = x 8 and g(x)= square root of x-1 (a) Find (f/g)(5). Show work. (b) Find the domain of the quotient function (f/g) Explain

Solution

(a) (f/g)(x) = f(x)/g(x)=(x-8)/(x - 1)

=>(f/g)(5)=(5-8)/(5 - 1)=-3/2

(b) Since (f/g)(x) = f(x)/g(x) for x to be in the domain of (f/g)(x) it must be in the domain of f and in the domain of g. You also need to insure that g(x) is not zero since f(x) is divided by g(x). Thus there are 3 conditions.

x must be in the domain of f: f(x) = x -8 and all real numbers x are in the domain of x.

x must be in the domain of g: g(x) = (x - 1) so x - 1 0 so x 1.

g(x) can not be 0: g(x) = (x - 1) and (x - 1) = 0 gives x = 1 so x 1.

Hence to satisfy all three conditions x, x 1 and x 1 so the domain of (f/g)(x) is all x satisfying x > 1.