AII.7 - The student will investigate and analyze functions algebraically and graphically. Key concepts include a) domain and range, including limited and discontinuous domains and ranges; b) zeros; c) x - and y -intercepts; d) intervals in which a function is increasing or decreasing; e) asymptotes; f) end behavior; g) inverse of a function; and h) composition of multiple functions. Graphing calculators will be used as a tool to assist in investigation of functions.

 The domain of a function represents all the x - values the function can assume.  When finding the domain, we look at what values the function uses as it goes left to right.  In Algebra 1 you studied linear (lines) and quadratic (parabolas) functions. Both of these functions continued to the left and right indefinitely (except for a vertical line), and thus they have an infinite domain. We say the domain of such functions is ‘all real numbers’.

 We can mathematically represent ‘all real numbers’ in different ways:  Set notation: { x | } ◦ Read as ‘ x such that x is an element of the set of real numbers’ ◦ This is the preferred notation for the SOL test.  Interval notation: (-∞, ∞) ◦ Read as ‘negative infinity to positive infinity’

 Which of the following functions has a domain of ‘all real numbers’? Yes! No

 This year you will learn about all the reasons that domains might be restricted – ◦ Asymptotes ◦ Holes ◦ Function limitations (like radicals)  But until you learn about all that, the best way to determine the domain of function is to see its graph.  If you are given the equation, use your calculator to get the graph.

Does the graph go to the right forever? Yes! Does the graph go to the left forever? No! The graph stops at (0, 3) How far to the left does the graph go? Remember, we are looking at the x -values. x = 0 For this function, the domain is any value greater than or equal to 0. Domain: { x | x ≥ 0}

Does the graph go to the left forever? How far to the left does the graph go? Remember, we are looking at the x -values. It looks like it is going to x = 0, but the graph turns and becomes almost vertical. So the function never actually gets to 0. For this function, the domain is any value greater than 0. Domain: { x | x > 0} Does the graph go to the right forever? Yes! No! This arrow points down.

Does the graph go to the left forever? Normally we would say the domain is ‘all real numbers’. But there is one more thing to check: Are there any missing points or breaks in the graph? Yes! There are three points of discontinuity: x = -2, x = -1, and x = 1. The domain is any value other than x = ± 1. Domain: { x | x ≠ ±1} Does the graph go to the right forever? Yes! But one of them gets filled in: x = -2 is used by the ordered pair (-2, 3). So we must include -2 in our domain.

 The range of a function tells us all the y -values the function can assume.  When finding the range, we look at what values the function is using as it goes up and down.  It is much more common for the functions we study to have limited ranges than limited domains.

 Which of the following functions has a range of ‘all real numbers’? No Yes! No Yes!

Does the graph go to up forever? Yes! Though function is not very steep, it is inching up as it goes to the right. Does the graph go down forever? No! The graph stops at (0, 3) How far down does the graph go? Remember, we are looking at the y -values. y = 3 The range is any value greater than or equal to 3. Range: { y | y ≥ 3}

Does the graph go down forever? Do we use all the values in between? Yes! The range is any value greater than or equal to 0. Range: { y | y ≥ 0} Does the graph go up forever? No! The lowest point is (0, 0). Yes!

Does the graph go down forever? Do we use all the values in between? Yes! The range is any value less than or equal to 1. Range: { y | y ≤ 1} Does the graph go up forever? No! The highest point is (0, 1). Yes! This image does not have arrows but we know parabolas continue indefinitely..

 Give the domain and range of each function. D: R: R: y ≥ 3 D: x > 0 R: y ≤ 1 D: x ≠ ±1 D: x ≥ 0 R: y ≥ 0