Question

How do you find the domain of $f + g$ given $f(x) = 3x + 4$ and $g(x) = \dfrac{5}{{(4 - x)}}?$

Answer 1

As we know that the domain of a function refers to the set of all possible input values that are present in the $x -$axis. It is the sum of two functions which is the intersection of their domains. Here in this question we will first find the domain of $f(x)$ and $g(x)$ independently. A function with a fraction with a variable in the denominator, to find the domain of this type of function we set the bottom part equal to zero and exclude the $x$ value and solve it.

Complete step by step solution:
Here in the above question $f(x)$ is a polynomial function, so its domain is $R$. And $g(x)$ is a fractional function, so its domain is $R$ except those points where the denominator vanishes.
Now equating the denominator to zero we get, $(4 - x) = 0$, so we get $x = 4$.
So the domain of both the function according to the question: $g(x)$ is $R - - \\{ 4\\}$, Now the domain of $(f + g)(x) = 3x + 4 + \dfrac{5}{{4 - x}}$, These consists of points where both $f(x)$ and $g(x)$ exist, this is called the intersection of both the domains. Since the domains of both are $R$ except the second one, which excludes the value i.e. $x = 4$, so the domain of the sum is : $Dom(f + g)(x) = R - \\{ 4\\}$.
Hence the required answer is $Dom(f + g) = R - \\{ 4\\}$.

Note: We have to be careful when solving this type of question. Whenever we get this type of question we first need to find the domains of individual functions and then solve further to get the required domain. Domain is an independent variable, for calculating domain we must check that if there is diffraction in the question then the denominator cannot be zero.