Although I technically only finished the unit test this morning (Monday, Nov. 25^th), I managed to get through the unit Functions this past week. This unit has been the most difficult for me in the course Algebra 1.

I began the week working through questions on intervals while using number lines and learned what the differences between closed and open intervals are:

• Closed Interval
  • Ex. -2 is less-than-or-equal-to X, X is less-than-or-equal-to 3
  • [-2, 3]
  • Square brackets (“[“ and “]”) are used to denote a closed interval.
  • On a number line, a closed interval uses a filled in circle.
  • A closed interval includes the number shown, i.e. -2 and -3 in the example above WOULD be apart of the interval.
• Open Interval
  • Ex. -1 < X < 3
  • (-1, 3)
  • Rounded brackets (“(“ and “)”) are used to denote a open interval.
  •  On a number line, an open interval uses a open in circle.
  • A open interval excludes the number shown, i.e. -2 and -3 in the example above would NOT be apart of the interval.

I then worked on Interval Notation which is a fancy way to say the written equation used to express an interval. Here I learned the meaning or the different symbols used, all of which I had never seen before:

• “{” and “}” brackets mean “set of values”
• A symbol that looks like ”E” but has a rounded back means “member of”
• A symbol that looks like ”R” but has two lines on the left edge means “real number”
• “|” means “such that” (Note: this is not a lower case L, but a vertical line/dash)
• Ex. of this notation:
  • {XER|XE[-3,2]}
  • “In a given set of values ({ and }), X is a member (E) of the Real numbers (R) such that (|) X is a member (E) of a closed interval ([ and ]) between -3 and -2”

After that I worked through questions that gave me a better understanding of the Domain and Range of functions. I learned that:

• Domain = All values that can be used in a function based on the definition/rules of the function, and
• Range = All set of values that can be outputted from a function based on the functions definition/rules.

I was also reintroduced to the terms “Defined” and “Undefined” numbers. These came up because, when looking at the formula of a given function, it’s important to know which numbers can be used in the formula (i.e. what inputs would result in a defined number as the outcome) and which numbers can’t be used (i.e. what inputs would result in an undefined number as the outcome). Notable examples of Defined and Undefined numbers include:

• Defined:
  • 0/2 or 0/x
  • The square root of a positive number
• Undefined
  • 2/0 or x/0
  • The square root of a negative number

I learned that a function can only have one output for any given input. This means that any number inside a functions domain can only output one number in its range. On a graph, this means a function cannot have two coordinates with the same x value and different y values. For Instance:

• The coordinates (2, 3) and (2, 4) would not be a function since there are two y values for the same x value.
• The coordinates (-4, 6) and (1, 6) could be potentially be apart of a function since, though they have the same y value, they do not have the same x value.

When looking at a graph of a function, I learned that highest and lowest points of the ‘peaks’ and ‘valleys’ have specific definitions:

• Absolute Maximum = The highest point of the range.
• Absolute Minimum = The lowest points of the range.
• Relative Maximum = The highest point on any ‘peak’ on the graph.
• Relative Minimum = The lowest point on any ‘valley’ on the graph.

The most challenging part of this unit came at the end when I was introduced to Inverse Functions. An inverse function is often denoted as f^-1(x) (f and x can be substituted with any variable, however) which when saying out loud is pronounced “f inverse of x”. My understanding is that an inverse function is the formula which essentially works backwards and uses the output produced by a function to find the corresponding input of that same function. Key things to remember about inverse functions include:

• On a graph, the line of an inverse function appears as a mirror image of it’s corresponding function.
• Flipping the coordinates of any given point within a function gives the coordinates of the counter point of the inverse function:
  • Ex. if the coordinates (-2, 4) fall on the line of a function, the coordinates (4, -2) would be coordinates of the inverse functions.

After having done KA for 12 weeks, I’ve realized that sometimes taking a break from certain concepts and coming back to them later helps for them to really sink in. I think this will likely be the case for functions. I’m still struggling to understand the language used when describing functions. Replacing y with f(x) it still a bit confusing to me. That said, it started making a bit more sense to me this morning so I think there’s a good chance that when I come back to functions it will start to become more clear.

This coming week I’m hoping to get through the units Sequences (0/1300 M.P. completed) and Absolute Value and Piecewise Functions (0/600 M.P. completed). I have a goal of finishing the Algebra 1 course by the New Year so, in order to do so, it’s important to get through these units in a timely manor and stay on track. As much as I’m enjoying getting better at math, I really don’t want to be ringing in the New Year sitting in front of my computer working on KA.