Algebra and Trigonometry, 5th Edition

2.2 The Algebra of Functions

• Find the sum, the difference, the product, and the quotient of two functions, and determine the domains of the resulting functions.

• Find the difference quotient for a function.

The Algebra of Functions: Sums, Differences, Products, and Quotients

We now use addition, subtraction, multiplication, and division to combine functions and obtain new functions.

Consider the following two functions f and g:

$$f(x)=x+2\text{and}g(x)=x^2+1.$$

Since $f(3)=3+2=5$ and $g(3)=3^2+1=10$, we have

$$f(3)+g(3)=5+10=15,$$

$$f(3)-g(3)=5-10=-5,$$

$$f(3)\cdot g(3)=5\cdot 10=50,$$

and

$$\frac{f(3)}{g(3)}=\frac{5}{10}=\frac{1}{2}.$$

In fact, so long as x is in the domain of both f and g, we can easily compute $f(x)+g(x)$, $f(x)-g(x)$, $f(x)\cdot g(x)$, and, assuming $g(x)\ne 0$, $f(x)/g(x)$. We use the notation shown below. ...