MTH 065 Class notes Lecture 3 (1.4b 1.6) Section 1.4: Linear Inequalities Graphing Linear Inequalities and Interval Notation We will look at two ways to write the solution set for a linear inequality: graphing the solution on a number line and writing it in interval notation. When you graph a linear inequality, you draw a number line and mark all the solutions. One end of the solution, the boundary, is marked with a parenthesis or bracket (depending on the inequality). Refer to the examples on top of page 34 for further illustration. Interval notation is similar to graphing; however you identify both ends of the solution set. One end is the number you already have, called the boundary. The other end is headed off to infinity in either the positive or negative direction. Then, just like with graphing, you use a parenthesis or bracket at the boundary. The end that is infinity always gets a parenthesis. Let s looking at some examples of interval notation: Note: The end with the infinity symbol always gets a parenthesis. The other end depends on the inequality symbol. Practice 4 Solve the inequality given. Then graph the solution on a number line, and write the solution set in interval notation. ( LCD: 2 OR Interval Notation: (
Checking Solutions to Linear Inequalities There are two parts to our answer that can be wrong: the boundary number and the inequality symbol. Practice 5 Solve the inequality Check your solution set. One possible solution: 0. We will plug this into our inequality and it should result in a true statement. True Section 1.5: Relations and Functions A relation as a rule that relates one set of numbers to another set of numbers. We call the first set of numbers the set of inputs and the second set of numbers, the set of outputs. A function is a relation where each input is assigned exactly one output. We call this relation 1 to 1. Your graphing (or scientific) calculator is another way we can generate a relation that is a function. You type in an input, you choose the rule and the calculator gives the output. Suppose you type in the number 5 for an input value. Now press the key x 2 (Pressing the key tells the calculator what rule to use to get the output.) The result, or output value, is 25.
Practice 1 Use the following table for A. x 3 6 7 2 0 y=x 2 9 36 49 4 0 A. How many values did you record in each cell in the table? One value B. Name at least two other function keys on your calculator. There are several other function keys on your calculator. Here are some examples: When working with relations, it is helpful to identify which variable is the input and which is the output. We do this by assigning the names independent and dependent to the variables. The variable, x, that we used in the table above to represent the input values is called the independent variable. The variable, y, that represents the output values is called the dependent variable. A way to remember this is the dependent variable depends upon both the input value and the rule to get its output value. Practice 2 For the relation given below, identify the independent and dependent variables. Jamie buys apples at a price of $1.29 a pound. The total price he pays is related to the number of pounds of apples he buys. Independent Variable: Number of pounds of apples Dependent Variable: Total cost of apples Domain and Range When working with relations we have two sets of numbers. The set of all input values is called the domain. The set of all output values is called the range. When looking at a graph, the domain is always associated with the horizontal axis and the range is associated with the vertical axis. It is common to use the graph of a function to determine what the domain and range are. When working with a table of values, unless otherwise indicated, the first row (or column) in a table contains the independent variable s values: the domain. The second row (or column) in the table contains the dependent variable s values: the range. Input Independent Variable Domain Horizontal Axis Output Dependent Variable Range Vertical Axis
Practice 3 The table defines a function between two variables. s -4-3 0 3 4 t -5-4 -1 2 3 (a) Identify the independent and dependent variables. Independent Variable: s Dependent Variable: t (b) Determine the domain and range. Domain: {-4, -3, 0, 3, 4} Range: {-5, -4, -1, 2, 3} Practice 4 Given the relations in the tables below, answer the following questions. A. x 0 1-1 2-2 y 1 2 2 5 5 B. t 0-1 -2 1-1 p 1-1 -1 2-2 (a) Does the table represent a function? Explain. A. Yes because for each input there is exactly one output. B. No because the input -1 does not have one output, -1 and -2. (b) What is the domain? A. {0, 1, -1, 2, -2} B. {0, -1, -2, 1} (c) What is the range? A. {1, 2, 5} B. {1, -1, 2, -2} Identifying Functions from a Graph We want to be able to tell from the graph whether a relation is a function or not. The definition of a function tells us that for each input value, we have exactly one output value. Graphically this means that two different points on the graph of a function can t have the same x-coordinate. A simple test to determine whether a relation is a function is called the vertical line test. The Vertical Line Test A relation is not a function if a vertical line could be drawn that intersects the graph in more than one point. Otherwise the relation is a function.
A. B. C. If it is possible to draw even one vertical line that intersects in more than one place, then the relation is not a function. So note that both Figure A and Figure B are not functions. However, figure 3 is a function because a vertical line placed anywhere on the graph will intersect the graph in exactly one point. Practice 5 For each of the following graphs, (1) Determine whether the graph represents a function. Explain. (2) Give the domain and range. A. B. (1) No this graph is not a function, does not pass vertical line test at. Note that this can be generalized that no vertical line can be a function. (2) Domain: {2} Range: (1) Yes, This graph does represent a function because a vertical line anywhere on the grid will intersect the graph in exactly one point. (2) Domain: Range: Practice 6 A. Heather is driving is driving south on Interstate 5. In 30 minutes she drove 35 miles. After 48 minutes she had driven 56 miles and in one hour she had driven 70 miles. Is the distance Heather drove a function of the time she had been driving? Explain. Time (min.) 30 48 60 Distance (mil.) 35 56 70 Yes, every input has one output
B. Jeremy, Chris and Tom all went to the post office today to buy some stamps. Jeremy bought 20 stamps and paid $7.80. Chris bought 20 stamps and paid $9.00. Tom bought eight stamps and paid $1.92. Is the amount they paid for stamps a function of the number of stamps? Explain. $6.00 C o s t ( D o l l a r s ) $5.00 $4.00 $3.00 $2.00 $1.00 $0.00 0 1 2 3 4 5 Number of stamps No because our graph does not pass the vertical line test. Increasing, Decreasing, and Constant Functions As the value of the independent variable increases, if the value of the dependent variable: always increases, the function is increasing. always decreases, the function is decreasing. remains the same, the function is constant. If we were to look at the graph, then Look at the graph from left to right. Is it going up? (Increasing) Is it going down? (Decreasing) Is it a horizontal line? (Constant) Practice 7 Determine whether the relation is increasing, decreasing, constant, or a combination. Increasing; as x increases, y also increases.
Section 1.6: Function Notation and Linear Functions Function Notation Not every equation is a function (where each input has exactly one output) so it is helpful to have a specific notation for equations that are functions. The letter f (for function), is most commonly used when writing functions. Other commonly used letters are g, h, r and s but you can use any letter you like. If the independent variable is n, then the dependent variable would be written as. Function Notation means f is a function of n is read f of n or "f acting on n". One of the benefits of function notation is that you can tell which of the variables is the independent variable. The variable that appears inside the parentheses on the left side is the independent variable of the equation (input value). With an equation you don t know, unless it is specified. Practice 1 Write each equation in function notation. (a), where a is the independent variable. (b), where t is the independent variable. So, why use this notation? Although it appears easier to use the original equation, there are several reasons that function notation is useful. One simple reason is that when we see function notation being used, we know that the equation represents a function. We do not have to use the definition of a function to test it. Another convenience of function notation is that it provides a shorter way of saying that we want to evaluate a function for specific values. Practice 2 Given the function (a). Find each of the following: (b)
(c) In order for a graph to show every possible solution, we do not have to solve several different points. Once we see a trend in the data, we can generalize by connecting solutions with a solid line of points. Eventually this leads to an example like the one below. Practice 3 Jamie buys apples at a price of $1.29 a pound. The total price he pays is related to the number of pounds of apples he buys. An equation for the total price he pays is: where n is the number of pounds that Jamie buys and is the total price he pays. (a) Identify the independent and dependent variables. Independent Variable: n Dependent Variable: T(n) (b) Create a table of values for the function. n 0 1 T(n) 0 1.29 2 2.58 4 5.16 6 7.74 (c) Graph the function. $10.00 $8.00 $6.00 Cost T(n) $4.00 $2.00 $0.00 0 2 4 6 Number of pounds of apples (n) 8
Linear Functions For a function to be linear: Equal steps in the independent variable will always produce equal steps for the dependent variable. Equal steps means that the difference between any two pairs of consecutive entries in the table are equal. Let's look at two tables that represent relations and determine whether they are linear functions. Look for equal steps in the input and equal steps in the output. That makes it a linear function. Practice 4 Do the tables below represent linear functions? (a) x -1 0 1 2 3 y -5-3 -1 1 3 Yes because equal steps of 1 in x produced equal steps of 2 in y. (b) c -3 0 3 6 9 d 8 4 0 4 8 No, because equal steps of 3 in x did not produce equal steps in y.