Question

For the functions f(x)=x/(x-1) and g(x)=13/x, find the composition of (f o g) and simplify your answer as much as possible. Write the domain using interval notation.

Answer 1

The composition of the functions f(x) = x/(x-1) and g(x) = 13/x is (f ∘ g)(x) = 13 / (13 - x). we simplified and found the restricted domain. The domain using interval notation is (-∞, 0) ∪ (0, 13) ∪ (13, +∞).

Step-by-step explanation:

First, we find the composition of the functions f(x) = x/(x-1) and g(x) = 13/x to find (f ∘ g)(x). To do that, we substitute g(x) into f(x):

(f ∘ g)(x) = f(g(x)) = f(13/x) = (13/x) / ((13/x) - 1)

Now simplify:

(13/x) / ((13/x) - 1) = (13/x) / (13/x - x/x) = (13/x) / ((13-x)/x) = 13 / (13 - x)

Next, we consider the domain of (f ∘ g)(x). Both f(x) and g(x) are undefined when their denominators are zero. Therefore:

• For g(x), the domain excludes x = 0, since the denominator cannot be zero.
• For (f ∘ g)(x), the domain also excludes x = 13, to prevent division by zero.

The domain of (f ∘ g)(x) = 13 / (13 - x) is thus all real numbers except x = 0 and x = 13, which in interval notation is: (-∞, 0) ∪ (0, 13) ∪ (13, +∞).