# How do you find the domain of f+g given f(x)=3x + 4 and g(x) = 5/(4-x)?

Jun 15, 2016

The domain of the sum of two functions is the intersection of their domains.

#### Explanation:

First of all, let us find the domain of $f \left(x\right)$ and $g \left(x\right)$ independently:

• $f \left(x\right)$ is a polynomial function, so its domain is $\mathbb{R}$.
• $g \left(x\right)$ is a fractional function, so its domain is $\mathbb{R}$ excepting those points where the denominator vanishes:

$4 - x = 0 \rightarrow x = 4$

So the domain of $g \left(x\right)$ is $\mathbb{R} - \left\{4\right\}$.

Now, the domain of

$\left(f + g\right) \left(x\right) = 3 x + 4 + \frac{5}{4 - x}$

consists of those points where both $f \left(x\right)$ and $g \left(x\right)$ exist, this is the intersection of both domains. Since both domains are $\mathbb{R}$ except the second one, which excludes the value $x = 4$, the domain of the sum is:

$\text{Dom} \left(f + g\right) \left(x\right) = \mathbb{R} - \left\{4\right\}$