Find the Domain

Question

Let
f(x) =
1/
x − 3
and
g(x) =
1/
x − 4
.
Determine the domain of the composite function. (Enter your answer using interval notation.)
g compose f




Find the composite function.
(g compose f)(x)=

Answer 1

For g(f(x)) to be defined, x must be in the domain of f  [so that f(x) is defined], and f(x) must be in the domain of g [so that g(f(x)) makes sense].  

 

Domain f = all real numbers except 3   (so x≠ 3)

 

Domain g = all real numbers except 4  (so f(x) ≠ 4)

 

x ≠ 3 and 1/(x-3) ≠ 4

 

x ≠ 3 and x - 3 ≠ 1/4

 

x ≠ 3 and x ≠ 3.25

 

domain g(f(x)) = (-∞, 3) ∪ (3, 3.25) ∪ (3.25, ∞)

 

g(f(x)) = g(1/(x-3)) =          1            =     1               =     x - 3    

                                              1      - 4         1 - 4(x - 3)        13 - 4x

                                         x - 3                        x - 3

 

 

Answer 2

g(f(x)) =

 

g(1/(x - 3))  =

 

1 / ((1 / (x - 3))  - 4) =

 

1 / ((-4x + 13) / (x - 3)) =

 

(x - 3) / (-4x + 13)

 

 

We set the denominator equal to zero.

 

-4x + 13 = 0

-4x = -13

 

   x = 13/4

 

The domain is  (-∞, 13/4)∪(13/4 , ∞)