Functions PDF

Representation of Functions General Mathematics Semi Nazareno, LPT, MAEd Lesson Objectives At the end of the lesson, the students must be able to: define functions and related terms; determine if the given relation represents a function; define piece-wise function; and represent...

Representation of Functions General Mathematics Semi Nazareno, LPT, MAEd Lesson Objectives At the end of the lesson, the students must be able to: define functions and related terms; determine if the given relation represents a function; define piece-wise function; and represents real-life situations using functions, including piece-wise functions.  Relation A relation is a set of ordered pairs. The domain of a relation is the set of first coordinates. The range is the set of second coordinates. Example of Relations 1. {(1, 4), (2, 5), (3, 6), (4, 8)} 2. {(4, 2), (4, -2), (9, 3), (7,3)} 3. {(1, a), (1, b), (1, c), (1,d)}  ABSCISSA (2, 5) ORDINATE Functions A function is a relation in which each element of the domain corresponds to exactly one element of the range. Examples of Functions 1. {(1, 4), (2, 5), (3, 6), (4, 8)} 2. {(2, 1), (3, 1), (4, 1), (5,1)} ACTIVITY 1 Determine if the following relations represent a function. 1. {(q, 0), (w, 1), (e, 2), (t, 3)} 2. {(-1, -2), (0, -2), (1, -2), (2, -2)} 3. {(1, 0), (1,1), (1, 2), (1, -2)} 4. {(x, 3), (y, 4), (z, 3), (w, 4)}  Functions vs. Relations A "relation" is just a relationship between sets of information. A “function” is a well-behaved relation, that is, given a starting point we know exactly where to go.  Example People and their heights, i.e. the pairing of names and heights. We can think of this relation as ordered pair: (height, name) Or (name, height)  Example (continued) Name Height Joe=1 6’=6 Mike=2 5’9”=5.75 Rose=3 5’=5 Kiki=4 5’=5 Jim=5 6’6”=6.5 Jim Kiki Rose Mike Joe Joe Mike Rose Kiki Jim Both graphs are relations (height, name) is not well-behaved. Given a height there might be several names corresponding to that height. How do you know then where to go? For a relation to be a function, there must be exactly one y value that corresponds to a given x value.  Conclusion and Definition Not every relation is a function. Every function is a relation. Definition: Let X and Y be two nonempty sets. A function from X into Y is a relation that associates with each element of X exactly one element of Y.  Recall, the graph of (height, name): What happens at the height = 5?  Vertical-Line Test A set of points in the xy-plane is the graph of a function if and only if every vertical line intersects the graph in at most one point. NOT A FUNCTION FUNCTION MANY-TO-MANY MANY-TO-ONE NOT A FUNCTION NOT A FUNCTION FUNCTION FUNCTION  MANY-TO-MANY MANY-TO-ONE ONE-TO-ONE MANY-TO-ONE  MANY-TO-ONE ONE-TO-ONE  Representations of Functions Verbally Numerically, i.e. by a table Visually, i.e. by a graph, diagram Algebraically, i.e. by an explicit formula  Ones we have decided on the representation of a function, we ask the following question: What are the possible x-values (names of people from our example) and y-values (their corresponding heights) for our function we can have?  Recall, our example: the pairing of names and heights. x=name and y=height We can have many names for our x-value, but what about heights? For our y-values we should not have 0 feet or 11 feet, since both are impossible. Thus, our collection of heights will be greater than 0 and less that 11.  We should give a name to the collection of possible x-values (names in our example) And To the collection of their corresponding y-values (heights). Everything must have a name  Variable x is called independent variable Variable y is called dependent variable For convenience, we use f(x) instead of y. The ordered pair in new notation becomes: (x, y) = (x, f(x)) Y=f(x) (x, f(x)) x  Domain and Range Suppose, we are given a function from X into Y. Recall, for each element x in X there is exactly one corresponding element y=f(x) in Y. This element y=f(x) in Y we call the image of x. The domain of a function is the set X. That is a collection of all possible x-values. The range of a function is the set of all images as x varies throughout the domain.  Our Example Domain = {Joe, Mike, Rose, Kiki, Jim} Range = {6, 5.75, 5, 6.5}  More Examples Consider the following relation: Is this a function? What is domain and range? Visualizing domain of Visualizing range of  Domain = [0, ∞) Range = [0, ∞)  More Functions Consider a familiar function. Area of a circle: A(r) = πr2 What kind of function is this? Let’s see what happens if we graph A(r).  Graph of A(r) = πr2 A(r) r Is this a correct representation of the function for the area of a circle??????? Hint: Is domain of A(r) correct?  Closer look at A(r) = πr2 Can a circle have r ≤ 0 ? NOOOOOOOOOOOOO Can a circle have area equal to 0 ? NOOOOOOOOOOOOO  Domain and Range of A(r) = πr2 Domain = (0, ∞) Range = (0, ∞) Some Types of Functions Linear Function A function f is a linear function if f(x) = mx + b, where m and b are real numbers, and m and f(x) are not both equal to zero. Quadratic Function A quadratic function is any equation of the form f(x) = ax2+ bx + c where a, b, and c are real numbers and a ≠ 0. LINEAR FUNCTION QUADRATIC FUNCTION Some Types of Functions Constant Function A linear function f is a constant function if f(x) = mx + b, where m = 0 and b is any real number. Thus, f(x) = b. Identity Function A linear function f is an identity function if f(x) = mx + b, where m = 1 and b = 0. Thus, f(x) = x. Some Types of Functions Absolute Value Function The function f is an absolute value function if for all real numbers x, f(x) = x, for x ≥ 0 –x, for x ≤ 0 Piecewise Function A piecewise function or a compound function is a function defined by multiple sub-functions, where each sub-function applies to a certain interval of the main function's domain. ABSOLUTE VALUE FUNCTION Exercise A Determine whether or not each relation is a function. Give the domain and range of each relation. 1. {(2, 3), (4, 5), (6, 6)} 2. {(5, 1), (5, 2), (5, 3)} 3. {(6, 7), (6, 8), (7, 7), (7, 8)} Exercise B Tell whether the function described in each of the following is a linear function, a constant function, an identity function, an absolute value function, ora piecewise function. 1. f(x) = 3x − 7 2. g(x) = 12 3. f(x) = 3, if x > −5 -6, if x < −5 Exercise B Tell whether the function described in each of the following is a linear function, a constant function, an identity function, an absolute value function, or a piecewise function. 4. 5. DOMAIN AND RANGE ORDERED PAIRS {1, 2, 3, 4} {2, 5} {-1, 0, 1, 2, 4} {-5, -4, -3, -1, 3, 5} MAPPING DIAGRAM TABLE OF VALUES {-7, -4, -2, 8, 10, 19} {-19, -6, 1, 2, 15, 18} {-11, -5, 6, 9, 12, 17} {-20, -3, 4, 8, 9, 16} GRAPHS GRAPHS To find the domain, solve for y... To find the range, solve for x...  GRAPHS The domain of a function is the set of all the x-coordinates in the functions’ graph Domain {3 ≤ x ≤ 12} The range of a function is a set of all the y-coordinates in the functions’ graph. Range{6 ≤ y ≤ 12} Domain is {0 ≤ x ≤ 4} Range is {1 ≤ y ≤ 5} The graph shows the path of a golf ball. What is the domain of the given graph? A {0 < y < 100} B {0 ≤ y ≤ 100} C {0 ≤ x ≤ 5} D {0 ≤ x ≤ 5} What is the domain of this function? A {-1 ≤ x ≤ 5} B {-1 ≤ x ≤ 9} C {2 ≤ x ≤ 5} D {0 ≤ y ≤ 9} What is the domain of the function shown on the graph? A {-2 < y ≤ 2} B {-4 ≤ x ≤ 6} C {-4 < y ≤ 2} D {-2 < x ≤ 6} What is the range of the function shown on the graph? A {-4 ≤ y < 2} B {-4 ≤ x ≤ 6} C {-4 < y ≤ 2} D {-2 < x ≤ 6} Sometimes you will be asked to determine a REASONABLE domain or range The average daily high temperature for the month of May is represented by the function t = 0.2n + 80 Where n is the date of the month. May has 31 days. What is a reasonable estimate of the domain? Answer: {1 ≤ n ≤ 31} What is a reasonable estimate of the range ?  Our function rule is: t = 0.2n + 80 Our domain is {1 ≤ n ≤ 31} Our smallest possible n is 1 Our largest possible n is 31 To find the range, substitute 1 into the equation and solve. Then substitute 31 into the equation and solve.  Our function rule is: t = 0.2n + 80 Substitute a 1 Substitute a 31 t = 0.2n + 80 t = 0.2n + 80 t = 0.2(1) + 80 t = 0.2(31) + 80 t = 0.2 + 80 t = 80.2 t = 6.2 + 80 t = 86.2 Reasonable range is {80.2 ≤ t ≤ 86.2}  Functional Notation …is an equation that is a function which may be expressed using functional notation. Function notation replaces the independent variable, y with either f(x), g(x), or h(x). f(x) is read as “f__of__x” Does not mean g(x) is read as “g__of__x” multiplication! h(x) is read as “h__of__x”  Instead of writing, y = x + 3, we replace the y with f(x) f (x) = x + 3 and is read as “f of x equals x+3” _________________________ Function Read as Notation f (x) = x2 f of x equals x2 f (x) = x f of x equals the absolute value of x f (x) = x f of x equals the square root of x f (x) = 3 + x2 f of x equals 3 + x2 Evaluating functions For the function f(x) = 2x + 6, the notation f(3) means that the variable x is replaced with the value of 3. f(x) = 2x + 6 f(3) = 2(3) + 6 f(3) = 12  Given f(x) = 3x – 4, evaluate f(4). f(4) = 3(4) - 4 f(4) = 12 - 4 f(4) = 8 Given g(x) = 2x – 10, evaluate g(5). g(5) = 2(5) -10 g(5) = 10 - 10 g(5) = 0 Given f(x) = 4x + 8, determine the range given the domain of { 0, 2, 4 }. Range = { 8, 16, 24 } Evaluating Functions Given f(x) = 4x + 8, find each: f(2) = 4(2) + 8 = (8 + 8) f(−4a) = 4(-4a) + 8 = 16 = -16a+ 8 f(a+1) = 4(a + 1) + 8 = 4a + 4 + 8 = 4a + 12 Word Problem 1 To sell more T-shirts, the class needs to charge a lower price as indicated in the following table: The price for which you can sell x printed T- shirts is called the price function p(x). p(x) represents each data point in the table. Solution : Step 1: Find the slope m of the line using the slope formula m = y2 – y1 / x2 – x1 Step 2: Write the linear equation with two variables by substituting the values of m and (x1, y1) to the formula y – y1 = m(x – x1) —the point-slope form of a linear equation. y – y1 = m(x – x1) y – 540 = (-⅕) (x − 500) y – 540 = −(⅕)x + 100 y = − (⅕) x + 640 y = 640 – 0.2x Thus, the price function is p(x) = 640 – 0.2x. Word Problem 2 Find the dimensions of the largest rectangular garden that can be enclosed by 60 m of fencing. Solution Let x and y denote the lengths of the sides of the garden. Then the area A = xy must be given its maximum value. Express A in terms of a single variable, either x or y. The total perimeter is 60 meters. 2x + 2y = 60 x + y = 30 y = 30 – x Hence, A = xy A = x(30 – x) A= 30x – x2 Solution Write this equation in the vertex form by completing the square. A = –(x2 – 30x + 225) + 225 A = –(x – 15)2 + 225 The maximum area is 225 square meters. Since x = 15 (the width) and 30 – x = 15 (the length), the dimension that gives the maximum area is 15 meters by 15 meters. Problem 3 Sketch the graph of the given piecewise function. What is f(– 4)? What is f(2)? 𝑥 + 2, 𝑖𝑓 𝑥 ≥ 0 𝑓 𝑥 =ቊ 2 −𝑥 + 2, 𝑖𝑓 𝑥 < 0 Solution To the right of the y-axis, the graph is a line that has a slope of 1 and y-intercept of 2. To the left of the y-axis, the graph of the function is a parabola that opens downward and whose vertex is (0, 2). To sketch the graph of the function, you can lightly draw both graphs. Then darken the portion of the graph that represents the function. Solution To find the value of the function when x = – 4, use the second equation. f(– 4) = – (– 4)2+ 2 = – 16 + 2 = – 14 To find the value of the function when x = 2, use the first equation. f(2) = 2 + 2 = 4 Exercise C A zumba instructor charges according to the number of participants. If there are 15 participants or below, the instructor charges ₱500.00 for each participant per month. If the number of participants is between 15 and 30, he charges ₱400.00 for each participant per month. If there are 30 participants or more, he charges ₱350.00 for each participant per month. 1. Write the piecewise function that describes what the instructor charges. 2. Graph the function.