Chapter 1.2, Problem 5GCE

To determine

To write: conjectures about those functions which are sum, difference, product and quotient of two functions.

Answer to Problem 5GCE

The domains of $\left(f+g\right)\left(x\right)$ , $\left(f-g\right)\left(x\right)$ and $\left(f\cdot g\right)\left(x\right)$ consists of all the common inputs of both domains of $f\left(x\right)$ and $g\left(x\right)$ . The domains of $\left(\frac{f}{g}\right)\left(x\right)$ consists of all the common inputs of both domains $f\left(x\right)$ and $g\left(x\right)$ , but the denominator does not equal to zero.

Explanation of Solution

Given information:

The statement given is ‘functions which are sum, difference, product and quotient of two functions’.

Calculation:

Let the two functions be $f\left(x\right)$ and $g\left(x\right)$ respectively. The table of operation with function is as follows:

Operation	Mathematical expression for operation
Sum	$\left(f+g\right)\left(x\right)=f\left(x\right)+g\left(x\right)$
Difference	$\left(f-g\right)\left(x\right)=f\left(x\right)-g\left(x\right)$
Product	$\left(f\cdot g\right)\left(x\right)=f\left(x\right)\cdot g\left(x\right)$
Quotient	$\left(\frac{f}{g}\right)\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$

The domain of $f\left(x\right)$ and $g\left(x\right)$ is all real numbers. The domains of $\left(f+g\right)\left(x\right)$ , $\left(f-g\right)\left(x\right)$ and $\left(f\cdot g\right)\left(x\right)$ consists of all the common inputs of both domains of $f\left(x\right)$ and $g\left(x\right)$ , which is a real number but it must satisfy where the denominator does not equal to zero i.e. $g\left(x\right)\ne 0$

The domains of $\left(\frac{f}{g}\right)\left(x\right)$ consists of all the common inputs of both $f\left(x\right)$ and $g\left(x\right)$ domains but the denominator does not equal to zero i.e. $g\left(x\right)\ne 0$ . If the denominator is zero, it will make the expression undefined. So, those real numbers which makes the denominator zero will not be included in the domain of $\left(\frac{f}{g}\right)\left(x\right)$ .