Question

For the functions f(x)=3/x+4 and g(x)=7/x+1 , find the composition f × g and simplify your answer as much as possible. Write the domain using interval notation.(f ×g)(x)=domain of f×g :

Answer 1

In this case, we'll have to carry out several steps to find the solution.

Step 01:

Data

f (x) = 3 / x + 4

g (x) = 7 / x + 1

(f ° g) (x) = ?

Step 02:

Composition rational functions:

(f ° g) (x) =

`f\text{ (g(x)) = }(3)/(((7)/(x+1))+4)`
`f\text{ (g(x)) = }(3)/((7+4x+4)/(x+1))=\frac{3x\text{ +3}}{7+4x+4}`
`f(g(x))=(3x+3)/(4x+11)=(3(x+1))/(4x+11)`

Domain of (f ° g) (x) :

4x + 11 = 0

4x = -11

x = -11/4 ===> Undefined point (singularity)

Domain:

(-oo , -11 /4) U ( -11/4 , oo)

The answer is:

(f ° g) (x) = 3(x + 1) / 4x + 11

Domain : (-oo , -11 /4) U ( -11/4 , oo)