EXAMPLE A {(1, 2), (2, 4), (3, 6), (4, 8)}

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1 Name class date Understanding Relations and Functions A relation shows how one set of things is related to, or corresponds to, another set. For instance, the equation A 5 s shows how the area of a square is related to its side length. You can also think of an of the formulas in science as rules epressing how one quantit is related to another. A relation cannot alwas be summed up with a rule in the form of an equation. For instance, the following is a list of the amounts of mone Sarah has saved for a car after certain amounts of time: $0 after 1 month, $150 after months, $175 after 3 months, and $75 after months. This list epresses a relation between number of months and amounts of mone saved. Relations are a fundamental part of math, science, and making sense of numbers that come up in everda life. An situation that can be described with an equation, a set of coordinate pairs, a table of input and output values, or a graph of points on a coordinate plane involves a relation. Below are three was of epressing the same relation. Input Output 1 3 {(1, ), (, ), (3, ), (, )} One set of values in a relation is thought of as the inputs and the other as the outputs. The set of inputs is the domain of the relation, and the set of outputs is the range. EXAMPLE A Find the domain and range of the relation: {(0, 1), (0, ), (0, 3), (0, )}. Recall that each ordered pair is in the form (, ). The domain consists of the inputs, or set of -values. The onl input is 0. The range consists of the outputs, or set of -values. The outputs are 1,, 3,. Solution: The domain is {0} and the range is {1,, 3, }. 015 College Board. All rights reserved. 1 SpringBoard Algebra, Unit 1

2 A function is a special tpe of relation in which each element in the domain is paired with eactl one element in the range. All functions are relations, but not all relations are functions. EXAMPLE B Determine whether the relation is a function. Input Output The domain contains the input values, and the range contains the output values. The input value 1 is paired with onl one output value, 9. The input value 15 is paired with onl one output value, 5. The input value 17 is paired with onl one output value, 5. The input value 0 is paired with onl one output value,. Solution: The relation is a function. EXAMPLE C Determine whether the relation is a function. The domain contains the -values, 7, 1,, and. The range contains the -values,,, 5, and 7. The value 1 in the domain is paired with two values, and 7, in the range. So, it is not a function. Solution: The relation is not a function. 015 College Board. All rights reserved. SpringBoard Algebra, Unit 1

3 The vertical line test is a simple, visual wa to determine whether a graphed relation is a function. Vertical Line Test A relation is not a function if a vertical line intersects its graph at more than one point. EXAMPLE D Use the vertical line test to determine whether the relation is a function. Imagine a vertical line sliding along the curve from left to right. Because the line will not intersect the graph at more than one point at a time, the relation is a function. Solution: The relation is a function. 015 College Board. All rights reserved. 3 SpringBoard Algebra, Unit 1

4 EXAMPLE E Use the vertical line test to determine whether the relation is a function. Imagine a vertical line sliding along the graph from left to right. Because the line will intersect the graph at two points anwhere to the right of 5, the relation is not a function. Solution: The relation is not a function. 015 College Board. All rights reserved. SpringBoard Algebra, Unit 1

5 Function notation is often used to epress functions. In function notation, a function is written as f (), which is read as f of. In this notation, f is used to represent the function. f is not a variable and the parentheses do not signif multiplication. In the function f() 5 1 1, the values of are the inputs and the values of f() are the outputs, or the value of the function. To find the value of the function f() for a particular -value, substitute the given value of into the equation. EXAMPLE F Given f() 5 1 1, evaluate f(). Step 1: Substitute for in the function. f() 5 () 1 1 Step : Follow the order of operations and simplif Solution: f() 5 9 PRACTICE Find the domain and range of each. 1. Input Output {(, 3), (, 5), (, 7)} 015 College Board. All rights reserved. 5 SpringBoard Algebra, Unit 1

6 Determine whether each relation is a function.. Input Output {(, 3), (, 3), (, 7)} Find f (1) for each. 7. f () 5 3. f () f () The cost C to take students on a field trip is represented b the function C() If the total cost was $9, how man students went on the field trip? Show our work. 015 College Board. All rights reserved. SpringBoard Algebra, Unit 1

Student Page. Algebra/ Day #4 90 Minute Class Functions, Patterns and X-Y Tables

Student Page. Algebra/ Day #4 90 Minute Class Functions, Patterns and X-Y Tables Student Page Algebra/ Da #4 90 Minute Class Functions, Patterns and X-Y Tables Definition: A relation is an set of ordered pairs Ex: # {(,), (-7,6), (-,4)} # { (0,8), (-, ), (0,6)} Definition: A function