05.11.2019: Function Notation and Operations, 3/4: Products and Quotients of Functions

Today's soundtrack is Six Feet Under: Death Rituals, a plodding death metal album featuring Chris Barnes on vocals. He was a founding member of Cannibal Corpse, and thus a huge influence on modern death metal. I bought this album because the album that got me into death metal was Cannibal Corpse's "The Bleeding," and I loved Barnes' raw, gutteral vocal style on it.

This morning, I'm learning about multiplying and dividing functions.

As we discussed yesterday, we can multiply functions together to create a new function. Multiplication of functions can be expressed as either f(x)g(x) = h(x) or as (fg)(x) = h(x). Either one means the same thing: multiply f(x) by g(x) to get a new function, h(x). When we multiply functions, the domain and range of the new function are those that were common to the original functions.

Again, we can also divide functions by each other. This can be expressed as either (f/g)(x) = h(x) or as f(x)/g(x) = h(x). Either way, we are being asked to divide one function by the other to get a new function. When dividing functions, we should follow these steps:

How to Divide Functions

  1. Factor completely

  2. Cancel like terms

  3. Simplify the expression

  4. Identify the restrictions

  • All factors that were in the denominator prior to canceling like terms cannot equal 0

  • Identify any asymptotes by substituting very large positive numbers and very small negative numbers in for x

  • Determine whether any points of discontinuity exist

  • Identify common factors in numerator and denominator prior to canceling like terms

  • If any common factors exist, solve for x to determine the x-value of the point of discontinuity

  • Substitute the x-value into the simplified fraction, then solve for y to get the y-coordinate

  • List that y-coordinate in the restrictions

That's it for today! Next time, we'll be learning about composition of functions.

#SixFeetUnder #CannibalCorpse