Finding the domain of these functions?

Question

Let f(x) = 1/(x + 4) and g(x) = 8x/(x + 4) 

Find f (x) + g(x), f (x) − g(x), f(x)g(x), and f (x)/g(x) 

Find the intervals for the domain for each function and enter the endpoint values 
f (x) + g(x) = 

f (x) − g(x) = 

f (x)g(x) = 

(f (x))/(g(x)) = 

Thanks

Answer 1

The functions are f(x) = 1/(x + 4) and g(x) = 8x /(x + 4)

f(x) + g(x) = [1/(x + 4)] + [8x /(x + 4)]

f(x) + g(x) = (1 + 8x)/(x + 4)

 

f(x) - g(x) = [1/(x + 4)] - [8x /(x + 4)]

f(x) - g(x) = (1 - 8x)/(x + 4)

 

f(x)g(x) = [1/(x + 4)][8x /(x + 4)]

f(x)g(x) = 8x/(x + 4)²

 

f(x)/g(x) = [1/(x + 4)]/[8x /(x + 4)]

f(x)/g(x) = 1/8x.

Answer 2

f(x) + g(x) = (1 + 8x)/(x + 4)

The above function is rational function.

We know that all possible values of x  is domain of a function.

A rational function is simply fraction and in a fraction the denominator cannot be equal to 0 because it would be undefined.

To find which number make the fraction undefined create an equation where the denominator is not equal to zero.

x + 4 ≠ 0

x  ≠ - 4

So the domain of the function all real numbers except -4.

Domain set is {x | x ∈ R, x ≠ -4}

Domain of the function f(x) + g(x) in interval notation (-∞, -4) U ( -4,∞) .

 

Similarly

f(x) - g(x) = (1 - 8x)/(x + 4)

Domain of the function f(x) - g(x) in interval notation (-∞, -4) U ( -4,∞) .

 

f(x)g(x) = 8x/(x + 4)²

Domain of the function f(x)g(x) in interval notation (-∞, -4) U ( -4,∞) .

 

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

The above function is undefined at x = 0

So the domain of the function all real numbers except 0.

Domain set is {x | x ∈ R, x ≠ 0}

Domain of the function f(x)/g(x) in interval notation (-∞, 0) U ( 0,∞) .