Algebra of Functions - Formulas and Domain of a Function With Examples

A function in mathematics refers to a polynomial expression that gives us the relationship between two variables such that one of them is the independent variable with whom the value of the dependable variable is related. Arithmetic operations can be easily implemented on a function with only one required criterion. Whenever mathematical operations are performed on a particular function their domain must be the same. The domain of any algebraic function is the set of numbers which can be used as an input for that particular function or the set of numerical values for which the function holds true. 

The Algebra of functions essentially deals with the four main arithmetic operations that are, addition, subtraction, multiplication and division. As mentioned before, for all these operations to be implemented the domain of the two independent functions must be the same. The domain of the function that is the result of the mathematical operations lies at the point of intersection of domains of the two independent. 

How to Find the Domain of a Function Algebraically?

The domain of any function is the set of numbers that can be taken as input for the functions. In some cases, finding which numerical values do not satisfy the function can be an easier alternative to finding out the domain of the function rather than calculating the values for which the said expression holds true. These values either make the function undefined or pose a ‘break’ in the function. For example, when the denominator of your function is 0, the function becomes immediately undefined at that point. The easiest method of finding out the domain of a function is by tracing the graph of the function. At the same time, there are certain methods that can be implemented to easily the domain of function algebraically.

• Dividing the Function by 0: When a function is divided by 0 that is the denominator becomes zero for a function, and the function becomes undefined. Therefore all the values that give a 0 in the denominator will not be a part of the domain of the function. All other values except for those fall within the domain of the said function. 

Example: Let us take the function $f(x)=\dfrac{2}{x-8}$.

 When x – 8 = 0 , the function will be undefined. 

Thus when x = 8, the function does not hold true and this shows that 8 does not fall in the domain of the function. 

• Square Root of a Negative Real Number: This method works best for real numbers and cannot be extensively used for complex values. As per the theory, the square root of a negative real number is undefined (Since the square of any value, positive or negative is always positive). Thus any value that gives us the square root of a negative real number does not fall within the domain of the function. 

Example: Let us take the function $f(x)=\sqrt{x+8}$

At x + 8 < 0 , x < - 8. 

This means that when the value of x is less than that of -8, the function becomes a square root of a negative real number (here, -8). Thus any value less than - 8 will not be satisfying the above function and will therefore not fall within the domain of f(x). 

Important Relations and Functions Formulas Associated with the Algebra of Functions

There are certain important relations and function formulas associated with the algebra of functions. These are the rules that any mathematical operations performed on the said function must follow. As mentioned before, identifying the domain of any two functions is crucial before you start your arithmetic operations because these can only be implemented when the domain of the individual or independent functions are one and the same. The following list gives us the four most commonly used formulas in the field of the algebra of function

• Addition of function: (f + g)(x) = f(x) + g(x)

• Subtraction of functions: (f – g)(x) = f(x) – g(x)

• Multiplication of functions: (f • g)(x) = f(x) •g(x)

• Division of functions: $\left(\dfrac{f}{g}\right)(x)=\dfrac{f(x)}{g(x)}$, Where $g (x) \neq 0$

Here we have taken two functions in x, f(x) and g(x) such that they both have the same domain. The domain of the resultant functions as an outcome of the mathematical operations lies at the point of intersection of their individual domain.

Algebraic Function Examples

After the detailed discussion related to the algebra of functions, function and relations formulas and the domain of a function, we consider the following algebra examples to help you further understand the topic. 

Example 1: If f(x)=x+7 and g(x)=x²+5x-6 find the value of (f + g)(x).

Solution:

According to the formula, (f + g)(x) = f(x) + g(x)

Given, f(x)=x+7 and g(x)=x²+5x-6.

Therefore,(f+g)(x)=f(x)+g(x)

= (x+7) + (x² + 5x - 6) 

=  x² + 6x + 1

The required addition of the two functions gives us the new function.

(f + g)(x) =  x² + 6x + 1

Example 2: If f(x)=x–1 and g(x)=x²+5x-6, then find the value and domain of $\left(\dfrac{f}{g}\right)(x)$ 

Solution: 

As per the formula of the division of algebra of functions,

$\begin{align} &\left(\dfrac{f}{g}\right)(x)=\dfrac{f(x)}{g(x))}=\dfrac{x-1}{x^{2}+5 x-6} \\ &=\dfrac{x-1}{x^{2}+5 x-6} \\ &=\dfrac{x-1}{(x-1)(x+6)} \end{align}$

To find out the domain we take the first method and see for what values of x the function gives 0 in the denominator.

As we can see , $x – 1 \neq 0$ and $x + 6 \neq 0$

i.e., $x \neq 1$ and $x \neq –6$ 

This means that the domain of the function has all the real numbers except for -6 and 1.  

Therefore the required domain of the function is R – {-6,1}. 

Conclusion

Algebra of functions deals with the arithmetic operations that can be implemented on a system of functional equations if and only if the domains of the individual and independent functions are one and the same. The domain of the resultant polynomial expression that we get as the outcome of the arithmetic operations lies at the point of intersection of the domains of the two independent functions.