This week was pretty typical. I didn’t get as far as I hoped I would but feel good about my effort having spent ~6.5-7 hours on KA. I finished the unit Polynomials on Friday and got through a few videos in the first section of Composite Functions on Saturday. I spent most of this week solving equations which was much more enjoyable than studying theory and/or memorizing definitions.  I definitely got better and more confident at working with polynomials but I still feel like there’s plenty left to learn about them. That said, I’m much less intimidated by large polynomials than I was at the beginning of the week which is a good confidence booster going into calculus. 💪🏼 💪🏼 💪🏼

The first thing I went over this week were two words used when dealing with polynomials which were ‘roots‘ and ‘solutions‘. I had seen these words used many times before but, in retrospect, I realize I never had a solid understanding of what each word meant:

• Roots
  • Any value that can be inputted into a given function for x-coordinate to make y = 0.
  • Example:
    • f(x) = (x + 2)(x – 4)
      • Roots = (–2), (+4)
• Solutions 
  • When two functions cross over each other, the x-coordinate of where they cross is a ‘solution’ to both functions.
  • Example:
    • In the photo just below, the solution to the equation 2^x – 3 = (x – 6)^2 – 4 is 3 because that’s the x-coordinate that satisfies both functions, a.k.a. an x-coordinate that is a part of both functions.

After going through those two definitions, I spent the next few days working on polynomial long division. I got pretty good at going through this process but I still find it a bit confusing and can’t visualize what’s going on in my head. Here is an example question from my notes that explains the process to solve these types of equations:

I understand the polynomial long division process pretty well and don’t have too much difficulty with it but, as I said above, I can’t visualize what’s going on when you divide polynomials into each other. I can’t understand how two functions drawn on a graph would divide into each other. In any case, I understood them well enough to pass the unit test on my first try getting 9/9 questions correct.

Although I only got through 1 video and 2 exercises in the unit Composite Functions, I feel like I managed to get a good understanding for what a composite function is which is more-or-less a function within a function (or as I like to think of it, function-inception). Take for example the following photo of 3 functions:

A composite function could be h(g(f(x))) which would be pronounced “h of g of f of x.” To solve a composite function, you start on the inside of the function by figuring out what f(x) equals, then inputting that value into g to get another output, then putting that final output into h to come up with the answer. Here’s how it works when x = 2:

• h(g(f(2))) = h(g(2^2 – 1))
  • = h(g(3))
  • = h(4)
    • (Side note: from the picture you can see on function g’s coordinate table that g(3) = 4.)
  • = -1
    • (Side note: on the graph of function h you can see that h(4) = -1.)

Here’s a screen shot I took from KA that gives a visual representation of how this works and another way to write composite functions:

This coming week I’m hoping to get through this unit, Composite Functions (0/700 M.P.), by Friday at the latest. My score for the following unit, Trigonometry (600/600 M.P.), is already at 100% but I think it would be a good idea for me to go back and review most of the unit. There are sections in it that deal with trigonometric identities which I don’t remember much about and which would probably be useful for me to review before getting deeper into calculus. As usual, I’m hoping that when I go back through trig all of the things I used to find confusing will somehow magically be easy for me to understand. 🤞🏼