Open and Closed Intervals. For interval notation: We use brackets If an endpoint value is included, and parentheses if it isn t included.

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1 Lesson 4 Open and Closed Intervals Review: Shormann Algebra, Lessons 1, 14, 15, 0, 6 Rules For interval notation: We use brackets If an endpoint value is included, and parentheses if it isn t included. Definitions - Interval notation: An efficient method of describing a set of real numbers, like a set of x-values, or the domain of a function. Instead of using set-builder notation or inequality symbols, brackets and parentheses are used instead. Examples of open, closed, and partially closed intervals are shown below: OPEN INTERVAL CLOSED INTERVAL PARTIALLY CLOSED INTERVALS -5 < x < -5 x -5 x < -5 < x (-5, ) [-5,] [-5,) (-5,] As the definition says, interval notation is a way to efficiently describe a set of real numbers. Rather than using set-builder notation (Lesson 6), or a series of inequality symbols as shown above, brackets and parentheses are used along with end values in the set. It s also important to notice the connection between interval notation and infinitesimals (Lesson 0). Look at the examples above, do you notice how the difference between ) and ] is an infinitesimal? For the in examples above, the open interval is describing values that are an infinitesimal away from, while the closed interval includes. Recognizing the infinitesimal will help you discern the differences between open and closed intervals. Example 4.1 Use interval notation to designate the following intervals. Identify each interval as opened, closed, or partially closed. a) - < x 1 b) x 14 c) x! x < 10 { } solution: One way to know when to use brackets is if you see an equals sign somewh were in the expression. A quick observation should reveal to you that a) and b) will use brackets: a) (-,1] This is a partially closed interval, since 1 is included, but - isn t.

2 b) [14, ) This is a partially closed interval. Using interval notation, you always have to have two values, even though one of them, infinity, isn t really an endpoint, because infinity is a value that is increasing without bound. Therfore, it can t end, so we use parentheses to show it is open. c) (-,10) This is an open interval, since both sides contain parentheses. Example 4. Which of the following subsets of real numbers best describes a closed interval? A) [1,5] B) (1,5] C) (1,5) D) [1,5) solution: If it s a closed interval, it will have brackets on both sides. Therefore, Choice A is the correct choice. Example 4.3 Which of the following subsets of real numbers best describes the solution set for the inequality shown? x A) 0, 3 B),3 C),3 D) 3,0 solution: First, solve the inequality for x: x 3 x 3 x would start, or be closed at x = 3, but would include all real numbers less than 3, leaving it open to the left. Therefore, Choice B is correct. Example 4.4 Use interval notation to describe the domain of the following graphs. f(x) f(x) a) b) solution: These are actually the same graphs from Example 15.. Instead of using inequality symbols, we will use the more efficient interval notation to write the domains. Note also how an open interval relates to an open circle on a function s graph, and a closed interval relates to a closed circle. a) (-4,4] b) (-3, 5]

3 Practice Set 4 (subscripts tell you which lesson each problem came from) Use your best judgment as to when you should use a calculator. Use 3.14 for π Use interval notation to designate the following interval. 7 x Which of the following subsets of real numbers best describes what is referred to as a partially closed interval? A) (0,6) B) (0, ) C) [-,3] D) (-,8] If 10 6x = 400, find x If 3 x+4 = 9 x-1, find x The fly population started with males and females, and triples fortnightly (every weeks). How many flies are formed between the 3rd and 4th fortnight? Which of Euclid s propositions requires you to construct a perpendicular bisector as part of the construction process? A) Prop. 1 B) Prop. C) Prop. 3 D) Prop. 4 E) Prop Which of the following is the graph of the inverse of A) B) x y = 3? C) D) The following is the graph of which function? A) f (x) = (x 4) 3 B) f (x) = x C) f (x) = x D)f (x) = x

4 9 38. Find the volume of the pyramid shown. The base is a square. Round to 1 d.p. 10 in 1 in 11 in Solve for h. x 4 h = t In the following equation, a, b, and c are constants. If the equation is true for all values of x, what is the value of a? x(4x +5) + 3(4x + 5) = ax + bx + c Use the graph of g(x) to find g(g()). g(x) x Use similar triangles to estimate the width, in yards, of the sinkhole. Round to 1 d.p W (Mendel s Accountant) The population genetics software was used to create a model that showed the mean fitness (f) of a population decreased linearly each generation (g). The fitness was 0.96 after 5 generations, and 0.88 after 100 generations. Use this data to create a function describing fitness as a function of generation, f(g). What was the fitness after 60 generations? 15 3,34. (SAT) If x >, which of the following is equivalent to 1 1 x ? x + A) x + 3 B) x + 3x + C) x + 3 x + 3x + D) x + 3x + x + 3

5 16 10,9. Given a convex polygon of 5 sides, write a -column proof to show the sum of the interior angles = 540. Refer to Lesson 9 Rules on polygons in your proof The diagonals of a are perpendicular bisectors If you used the completing the square method to find the roots of y = 3x + 9x - 15, the squared binomial you create would equal which of the following? A) (3x + ) B) x + 3 C) x + 3 D) ( x + 6) Solve the following nonlinear system Factor. (x - 5) + k(x - 5) y = x + 4x +1 y = 4x +10