Lesson 4.2

 Rachel’s parents keep track of her height as she gets older. They plot these values on a graph and connect the points with a smooth curve.  For every age you choose on the x-axis, there is only one height that pairs with it on the y-axis. That is, Rachel is only one height at any specific time during her life.

 A relation is any relationship between two variables.  A function is a special type of relation such that for every value of the independent variable, there is at most one value of the dependent variable.  If x is your independent variable, a function pairs at most one y with each x. You can say that Rachel’s height is a function of her age.

 You may remember the vertical line test from previous mathematics classes. It helps you determine whether or not a graph represents a function.  If no vertical line crosses the graph more than once, then the relation is a function.  Take a minute to think about how you could apply this technique to the graph of Rachel’s height and the two graphs at the right.

 Function notation emphasizes the dependent relationship between the variables that are used in a function.  The notation y=f(x) indicates that values of the dependent variable, y, are explicitly defined in terms of the independent variable, x, by the function f.  You read “y=f(x)” as “y equals f of x.”

 Graphs of functions and relations can be continuous, such as the graph of Rachel’s height, or they can be made up of discrete points, such as a graph of the maximum temperatures for each day of a month.  Although real-world data often have an identifiable pattern, a function does not necessarily need to have a rule that connects the two variables.

 Function f is defined by the equation.  Function g is defined by the graph at right.  Find these values: a)f (8) b)f (-7) c)g (1) d)g (-2) e)Find x when g (x) =0. To find when g(x)=0, find a place where the y value is zero. That is at (-2,0)

 Below are nine representations of relations. To Be or Not to Be (a Function) g. independent variable: the age of each student in your class dependent variable: the height of each student h. independent variable: an automobile in the state of Kentucky dependent variable: that automobile’s license plate number i. independent variable: the day of the year dependent variable: the time of sunset

g. independent variable: the age of each student in your class dependent variable: the height of each student h. independent variable: an automobile in the state of Kentucky dependent variable: that automobile’s license plate number i. independent variable: the day of the year dependent variable: the time of sunset Identify each relation that is also a function. For each relation that is not a function, explain why not. For each graph or table that represents a function in parts a–f, find the y-value when x =2, and find the x-value(s) when y =3. Write each answer in function notation using the letter of the subpart as the function name. For example, if graph a represents a function, a(2) =? and a(?)= 3.

 When you use function notation to refer to a function, you can use any letter you like.  For example, you might use ◦ y=h(x) if the function represents height, or ◦ y=p(x) if the function represents population.  Often in describing real-world situations, you use a letter that makes sense. However, to avoid confusion, you should avoid using the independent variable as the function name, as in y=x(x). Choose freely but choose wisely.

 When looking at real-world data, it is often hard to decide whether or not there is a functional relationship.  For example, if you measure the height of every student in your class and the weight of his or her backpack, you may collect a data set in which each student height is paired with only one backpack weight. ◦ But does that mean no two students of the same height could have backpacks of different weights? ◦ Does it mean you shouldn’t try to model the situation with a function?