VIEWS: 5 PAGES: 29 POSTED ON: 11/4/2012 Public Domain
Functions ƒ(x) Function Notations A relation is a pairing between two sets. A function is a relation in which each x-value has only one y-value Functions can be represented in many ways including tables, graphs and equations. Take for example the equation y = 2x - 3. This equation has an important characteristic. For each value of x, you find exactly one value of y. Relation: A relation is simply a set of ordered pairs. A relation can be any set of ordered pairs. No special rules need apply. The following is an example of a relation: {(1,2)(2,4)(3,5)(2,6)(1,-3)} The graph at the right shows that a vertical line may intersect more than one point in a relation. Function: A function is a set of ordered pairs in which each x-value has only ONE y- value associated with it. The relation we just discussed {(1,2),(2,4,)(3,5)(2,6)(1,-3)} is NOT a function because the x-value 2 is paired with a y-value of 4 and 6. Similarly, the x-value of 1 is paired with the y-value of 2 and -3 The previous relation can be altered to become a function by removing the ordered pairs where the x- value is used twice. Function: {(1,2)(2,4)(3,5)} The graph at the left shows that a vertical line intersects only ONE point in a function. This is called the vertical line test for functions. Function – an input-output relationship that has exactly one output for each input. Domain – the set of all input (x)values of a function. Range – The set of all output (y)values in a function. Function notation – the notation used to describe a function. Example f(x) is read “f of x.” f(1) is read “f of 1.” Linear function – a function whose graph is a straight line. To determine if a relationship is a function, verify that each input has exactly one output. Using tables is one way to verify functions. Look at the function below. Can you determine if it is a relation or a function? You can identify functions using tables or graphs. The graph below has more than one output for each input. Is this a function? A relation can be represented by a set of order pairs (x, y) . The first number, x, is a member of the domain and the second number, y, is a member of the range. Determine whether the order pairs make a function. {(-1, 7), (0, 3), (1, 5), (0, -3)} {(0, 2), (2, 4), (4, 8), (8, 10)} Vertical-Line Test for a function If no vertical line in the coordinate plane intersects a graph in more than one point, then the graph represents a function. (You can use a pencil held vertically to test) Evaluating Functions For the function y = 2x - 1, find f(0), f(2), and f(-1). y = 2x – 1 f(x) = 2x -1 Write in function notation. f(0) = 2(0) – 1 = -1 f(2) = 2(2) – 1 = 3 f(-1) = 2(-1) – 1 = -3 Find f(1), f(2), f(3), and f(4). Read the graph to find y for each x. f(x) = y f(1) = 8 f(2) = 10 f(3) = 12 f(4) = 14 Linear Functions Straight Lines Writing equations of functions Use the equation f(x) = mx + b. Find b (y-intercept) = -4 Locate a point on the line, such as (2, 0). Substitute the values into your equation. f(x) = mx + b 0 = m(2) – 4 0 = 2m – 4 0 + 4 = 2m -4 + 4 4 = 2m 2 2 m=2 f(x) = 2x - 4 Writing an equation using a table The y-intercept can be identified from the table, (0, 1) X Y Pick a point, (1, 3) and substitute your point and y-intercept into your equation. -2 -3 f(x) = mx + b -1 -1 3 = m(1) + 1 0 1 3=m+1 3–1=m+1–1 1 3 m=2 2 5 f(x) = 2x + 1 Physical Science The relationship between the two temperatures in the table are linear. Write a rule for Fahrenheit temperature as a function of Celsius temperature. f(x) = mx + b, where x is Celsius and y is Fahrenheit Temperature (°C) Temperature (°F) -10 14 0 32 25 77 50 122 100 212 Practice with Functions 1) Which of the relations below is a function? a) {(2, 3), (3, 4), (5, 1), (2, 4)} b) {(2, 3), (3, 4), (6, 2), (7, 3)} c) {(2, 3), ( 3, 4), (6, 2), (3, 3)} 2) Given the relation A = {(5, 2), (7, 4), (9, 10), (x, 5)}. Which of the following values for x will make relation A a function? a) 7 b) 9 c) 4 3) The following relation is a function. {(10, 12), (5, 3), (15, 10), (5, 6), (1, 0)} • True • False 4) Which of the relations below is a function? a) {(1, 1,), (2, 1), (3, 1), (4, 1), (5, 1)} b) {(2, 1), (2, 2), (2, 3), (2, 4), (2, 5)} c) {(0, 2), (0, 3), (0, 4), (0, 5), (0, 6)} 5) The graph of a relation is shown at the right. Is this relation a function? a) yes b) no c) Cannot be determine from a graph 6) Is the relation depicted in the table below a function? a) yes b) no c) cannot be determined from a table X 0 1 3 5 3 9 Y 8 9 10 6 10 7 7) The graph of a relation is shown below. Is the relation a function? a) yes b) no c) cannot be determined from a graph 8) Is the relation in the table below a function? a) yes b) no x -2 -1 0 1 2 3 y 5 5 5 5 5 5 9) The graph of a relation is shown below. Is the relation a function? a) yes b) no c) cannot be determined from a graph 10) The graph of a relation is shown below. Is the relation a function? a) yes b) no c) cannot be determined from a graph 11) Given f(x) = 3x + 7, find f(5). a) 15 b) 22 c) 42 12) Which graph represents a function?