05.10.2019: Function Notation and Operations, 2/4: Sum and Difference of Functions

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

#LacunaCoil #Evanescence