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; c) x - and y -intercepts; d) intervals in which a function is increasing or decreasing; and f) end behavior.

Relation: a set of ordered pairs Domain: the set of x-coordinates Range: the set of y-coordinates Function: relation with no repeated x-coordinates When writing the domain and range, do not repeat values.

Given the relation: {(2, -6), (1, 4), (2, 4), (0,0), (1, -6), (3, 0)} State the domain: D: {0,1, 2, 3} State the range: R: {-6, 0, 4} Function? No: (2, -6) and (2, 4) have the same x-value.

Relations can be represented in several ways: ordered pairs, table, graph, or mapping. We have already seen relations represented as ordered pairs.

{(3, 4), (7, 2), (0, -1), (-2, 2), (-5, 0), (3, 3)}

Create two ovals with the domain on the left and the range on the right. Elements are not repeated. Connect elements of the domain with the corresponding elements in the range by drawing an arrow.

{(2, -6), (1, 4), (2, 4), (0, 0), (1, -6), (3, 0)}

Vertical Line Test: If there are no vertical lines that intersect the graph of a relation at more than one point, the graph represents a function.

x y x y No D: {x|x ≥ 1} R: all reals No D: all reals R: all reals Ex3 Ex4

Yes D: all reals R: {y: y ≥ -6} No D: {2} R: all reals x y x y Ex5 Ex6

 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 | ℝ} ◦ Read as ‘ x such that x is an element of the set of real numbers’ ◦ This is the preferred notation for the SOL test.  ℝ ◦ Read as ‘the set of all real numbers’  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: ℝ R: { y: y ≥ 3} D: { x | x > 0} R: { y | y ≤ 1} D: { x| x ≠ ±1} D: { x: x ≥ 0} R: { y: y ≥ 0}

Use ≤ ◦ Use < If both ends of domain or range go to infinity, use x є ℝ or y є ℝ You don’t need to write. You may leave that side blank ∞ When stating the domain and range of a function, you have to look at the ends of the graph of your function in order to properly label the domain and range.

D: R: 1.Start with the domain- how far to the left does the function go? -5 2.What is the end marking? What inequality symbol will you use? Closed dot, so ≤ 3.How far to the right does the domain go? What symbol will we use? 3, closed dot so ≤ 4.Now to the range- What is the lowest point my function reaches? What symbol? -3, ≤ 5.What is the highest? Symbol? 4, ≤ {x|-5 ≤ x ≤ 3} {y|-3 ≤ y ≤ 4} Inequality (set) Notation Ex12:

◦ Use [ or ] Use ( or ) Use (- or ) ∞ ∞ End Marking Notation ∞ always gets parenthesis!!

D: R: 1.Start with the domain- how far to the left does the function go? -5 2.What is the end marking? What interval symbol will you use? Closed dot, so [ 3.How far to the right does the domain go? What symbol will we use? 3, closed dot so ] 4.Now to the range- What is the lowest point my function reaches? What symbol? -3, [ 5.What is the highest? Symbol? 4, ] [-5, 3] [-3, 4] Interval NotationEx12:

D: R: 1.Start with the domain- how far to the left does the function go? -8 2.What is the end marking? What inequality symbol will you use? Closed dot, so ≤ 3.How far to the right does the domain go? What symbol will we use? Infinity, no symbol 4.Now to the range- What is the lowest point my function reaches? What symbol? Negative infinity, no symbol 5.What is the highest? Symbol? 4, ≤ {x|-8 ≤ x} Right side blank {y|y ≤ 4} Left side blank Inequality Notation Ex13:

D: R: 1.Start with the domain- how far to the left does the function go? -8 2.What is the end marking? What inequality symbol will you use? Closed dot, so [ 3.How far to the right does the domain go? What symbol will we use? Infinity, ) 4.Now to the range- What is the lowest point my function reaches? What symbol? Negative infinity, ( 5.What is the highest? Symbol? 4, ] [-8, ∞) (- ∞, 4] Interval Notation Ex13:

D: R: D: R: Interval Inequality Ex14:

D: R: D: R: Interval Inequality Ex15:

The end behavior of a graph describes the far left and the far right portions of the graph. describes what happens as x gets bigger and bigger describes what happens to the values f(x) of a function f as x increases without bound ( x  ∞) a nd as x decreases without bound ( x  - ∞)

 1. Values approach a real number.  2. The values of f(x) increase or decrease without bound; that is ◦ f(x)  ∞ or f(x)  - ∞  3. The values of f(x) follow neither 1 or 2.  (You’ll see examples like this in trig.)

 Determine the end behavior of each function. Up to left As x → ∞, y → As x → -∞, y → Up to right ∞ ∞ Down to left As x → ∞, y → As x → -∞, y → Up to right -∞ ∞

 Determine the end behavior of each function. As x → ∞, y → As x → -∞, y → 1 1 Up to left As x → ∞, y → As x → -∞, y → Down to right ∞ -∞ 1

A function f is increasing on an interval if as x increases, f ( x ) increases. A function f is decreasing on an interval if as x increases, f ( x ) decreases. A function f is constant on an interval if as x increases, f ( x ) remains the same.

A function f is increasing on an interval if it goes up to the right. A function f is decreasing on an interval if it goes down to the right.

A function f is increasing on an interval if as x increases, then f ( x ) increases. A function f is decreasing on an interval if as x increases, then f ( x ) decreases. f ( x ) is increasing on the interval. vertex (1.5,-2) f ( x ) is decreasing on the interval.

A function f is increasing on an interval if as x increases, then f ( x ) increases. A function f is decreasing on an interval if as x increases, then f ( x ) decreases. Find the interval(s) over which the interval is increasing, decreasing and constant? Answer Now A function f is constant on an interval if as x increases, then f ( x ) remains the same.

f ( x ) is decreasing in the interval (-1,1). A function f is increasing on an interval if as x increases, then f ( x ) increases. A function f is decreasing on an interval if as x increases, then f ( x ) decreases. f ( x ) is increasing in the intervals A function f is constant on an interval if as x increases, then f ( x ) remains the same.

Find the interval(s) over which the interval is increasing, decreasing and constant? A function f is increasing on an interval if as x increases, then f ( x ) increases. A function f is decreasing on an interval if as x increases, then f ( x ) decreases. A function f is constant on an interval if as x increases, then f ( x ) remains the same. Answer Now

A function f is increasing on an interval if as x increases, then f ( x ) increases. A function f is decreasing on an interval if as x increases, then f ( x ) decreases. A function f is constant on an interval if as x increases, then f ( x ) remains the same. f(x) is decreasing over the intervals f(x) is increasing over the interval (3,5). f(x) is constant over the interval (-1,3).