# For the given functions f and g, find the following and state the domain of each result. f(x)=3x+1/4x-5; g(x)=5x/4x-5 For the given functions f and g, find the following and state the domain of...

For the given functions f and g, find the following and state the domain of each result.

f(x)=3x+1/4x-5; g(x)=5x/4x-5

For the given functions f and g, find the following and state the domain of each result.

f(x)=3x+1/4x-5; g(x)=5x/4x-5

What is the domain for f+g?

What is the domain of f/g?

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### 1 Answer

For the functions `f(x)={3x+1}/{4x-5}` and `{5x}/{4x-5}` , we can combine the functions in several ways. The domain of addition of two functions is the same as the union of the domains of the two functions.

In this case, the domain of both f(x) and g(x) is `{x in R|x ne 5/4}` since the domain cannot include the points where the denominator vanishes of the functions.

However, the domain of division of two functions is the union of the domain of the two functions and intersection with the domain of the resulting function.

With algebra, we see that `f/g={3x+1}/{4x-5} cdot {4x-5}/{5x}={3x+1}/{5x}` which has denominator vanishing at x=0.

**This means that the domain of f+g is `{x in R|x ne 5/4}`**

**and the domain of `f/g` is `{x in R|x ne 5/4, x ne 0}` .**