Today's soundtrack is *Lacuna Coil: Karmacode*, a metal album featuring Cristina Scabbia's gorgeous singing. I still remember discovering this band and thinking, "THIS is what Evanescence was going for, but couldn't quite reach!"

When we combine two or more functions, we get a *new function*!

Let's say we want to add *f*(*x*) =3*x*-1 to *g*(*x*)=*x*+2. So we are looking for (*f* + *g*)(*x*), the sum of *f* and g of *x*. Since 3*x* + *x* = 4*x*, and -1+2=1*, f*+*g*(*x*)=4*x*+1, but since combining two or more functions gives us a new function, our answer is *h*(*x*)=4*x*+1.

The four kinds of notations we will use for function operations are as follows:

Sum

(

*f*+*g*)(*x*) =*h*(*x*)Difference

(

*f*-*g*)(*x*) =*h*(*x*)Product

(

*fg*)(*x*) =*h*(*x*)Quotient

(

*f*/*g*)(*x*) =*h*(*x*),*g*(*x*)≠0

We will also work with *radical functions*, which are functions where the variable is the radicand. Whenever we come across these, we must bear in mind this restriction: All radicands must be greater than or equal to 0.

Finally, *rational functions* are functions with "fractions with variables" (www.contentconnections.ca). They take the form *f*(*x*)=P(*x*)/Q(*x*). When working with rational functions, remember the following tips about operating functions:

When multiplying, try to simplify before solving.

We can cancel out terms both diagonally and vertically when multiplying rational numbers.

When dividing fractions, we simply invert the fraction to the furthest right, then multiply. This is called "multiplying the reciprocal."

When adding or subtracting rational numbers, we must have a common denominator. The easiest way to find a common denominator if we have variables in the denominators is to multiply both the numerator and denominator of each fraction by the denominator of the other. This will give a common denominator and will allow us to add or subtract the two fractions before simplifying.

That's it for today. Next time, I'll be learning about products and quotients of functions.