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Intro to Functions
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 We’ll learn that a function is a special type of equation in which one input produces one output.
 We will also learn how to identify functions from graphs and data tables.
 And we’ll see that this knowledge can help us juice sugarcane, get carnival tokens, and make tasty treats!

Discussion Questions

Before VideoWhat is a relation?ANSWER

A relation is any rule that connects two variables. A relation can connect both quantitative and qualitative variables

A variable is a placeholder for an unknown measurement. Since measuring is the act of sizing attributes, a variable represents an unknown aspect of an attribute.

Relations are important because they help us describe relationships between two values or concepts. They allow us to define, discuss, and use relationships between variables in problemsolving.

I can represent a relation verbally, in a table of values, as a set of ordered pairs, and as an equation.

A linear relation forms a straight line. The equation of a linear relation has a rate and a constant, and it can be represented as y=mx+b.


After VideoWhat is the difference between a relation and a function?ANSWER

A function is a relation where each input produces only one output.

Function notation makes it easier to see the input and output in a function. Function notation says f(input)=output, which makes it easy to write the ordered pair (input, output), and easy to graph, since the input is on the xaxis and the output is on the yaxis.

A linear function has a graph that is a straight line. The function increases to the right if it has a positive slope (rate) and decreases if it has a negative slope (rate).

A function is nonlinear if its graph is not a straight line. Some examples of nonlinear functions are quadratic and cubic functions.

The vertical line test is a method for determining if a relation is a function using its graph on a coordinate plane. Imagine drawing a vertical line on the coordinate plane and scanning it from left to right. If at any point the vertical line would cross the relation in two or more places, the relation fails the vertical line test, so it is not a function.



Vocabulary

Function
DEFINE
A relation that takes an input and gives one output. We can represent functions with a table of values, ordered pairs, a graph, an equation, function notation, or words.

Input
DEFINE
A value that we substitute into a function that is transformed by the function into the output.

Output
DEFINE
The answer given by substituting the input into the function.

Vertical line test
DEFINE
A method for determining if the graph of a relation is also a function. If a vertical line scanned from left to right on the graph only intersects the relation once at any time, the relation is also a function. If at any xvalue the vertical line crosses the relation more than once, the relation is not a function.

Relation
DEFINE
Any rule that connects one variable to another. We can represent a relation with a table of values, a set of ordered pairs, a graph, an equation, or words.

Linear function
DEFINE
A function that, when graphed, forms a straight line.

Function
DEFINE

Reading Material
Download as PDF Download PDF View as Separate PageWHAT IS A FUNCTION?A relation shows a relationship between two quantities. A function is a special type of relation that, for each input, gives exactly one output.c definition here
To better understand functions…
WHAT IS A FUNCTION?. A relation shows a relationship between two quantities. A function is a special type of relation that, for each input, gives exactly one output.c definition here To better understand functions…LET’S BREAK IT DOWN!
Use an equation to represent a function for carnival tokens.
Emily wants to buy tokens for the carnival. The token machine says one dollar buys three tokens. A function is a special type of relation where one input gives one output. The machine works like a function, with an input of dollars and an output of tokens. You can represent the function mathematically using the equation y=3x, where x represents the number of dollars and y represents the number of tokens. If you want to know how many tokens she gets for $20, substitute 20 into the function for x and calculate the output number of tokens, y. For y=3x, y=320=60. So Emily gets 60 tokens if she puts $20 into the machine. Try this yourself: How many tokens does Emily get for $30?
Use an equation to represent a function for carnival tokens. Emily wants to buy tokens for the carnival. The token machine says one dollar buys three tokens. A function is a special type of relation where one input gives one output. The machine works like a function, with an input of dollars and an output of tokens. You can represent the function mathematically using the equation y=3x, where x represents the number of dollars and y represents the number of tokens. If you want to know how many tokens she gets for $20, substitute 20 into the function for x and calculate the output number of tokens, y. For y=3x, y=320=60. So Emily gets 60 tokens if she puts $20 into the machine. Try this yourself: How many tokens does Emily get for $30?Identify functions in lists and graphs.
You can use an equation to generate a set of ordered pairs for a function. For example, the equation y=3x generates the ordered pairs {(1, 3), (2, 6), (3, 9), (4, 12), (5, 15)...}. Notice that in the set of ordered pairs, no xvalues are repeated within the set, which tells us that it is a function. You can graph those points and connect them to show the graph of the function. In a function, each input (xvalue) can only produce one output (yvalue). If you draw a vertical line on the graph, then this vertical line only crosses the function once at any given xvalue. The Vertical Line Test is a method to determine if a graphed relation is also a function. To conduct the test, imagine scanning a vertical line from left to right on the graph of a relation. If at any point the vertical line crosses the function twice, the relation fails the test, and it is not a function. Try this yourself: Use a graphing calculator to graph a relation. Use the vertical line test to see if the relation is a function.
Identify functions in lists and graphs. You can use an equation to generate a set of ordered pairs for a function. For example, the equation y=3x generates the ordered pairs {(1, 3), (2, 6), (3, 9), (4, 12), (5, 15)...}. Notice that in the set of ordered pairs, no xvalues are repeated within the set, which tells us that it is a function. You can graph those points and connect them to show the graph of the function. In a function, each input (xvalue) can only produce one output (yvalue). If you draw a vertical line on the graph, then this vertical line only crosses the function once at any given xvalue. The Vertical Line Test is a method to determine if a graphed relation is also a function. To conduct the test, imagine scanning a vertical line from left to right on the graph of a relation. If at any point the vertical line crosses the function twice, the relation fails the test, and it is not a function. Try this yourself: Use a graphing calculator to graph a relation. Use the vertical line test to see if the relation is a function.Linear functions are relations with a rate and a constant.
Kenzie make $7 a week in allowance plus $2.50 per chore. You can represent the relation between the number of chores Kenzie does (x) and the number of dollars he earns in a week (y) with the equation y=2.5x+7. Each number of chores Kenzie does produces a different amount of allowance. So, this relation is also a function. This function has a rate ($2.50/chore) and a constant ($7). This function is a linear function. Try this yourself: Determine the amount of allowance Kenzie makes if he does 1 to 10 chores in a week, and then represent this function in a table of values, a set of ordered pairs, and a graph.
Linear functions are relations with a rate and a constant. Kenzie make $7 a week in allowance plus $2.50 per chore. You can represent the relation between the number of chores Kenzie does (x) and the number of dollars he earns in a week (y) with the equation y=2.5x+7. Each number of chores Kenzie does produces a different amount of allowance. So, this relation is also a function. This function has a rate ($2.50/chore) and a constant ($7). This function is a linear function. Try this yourself: Determine the amount of allowance Kenzie makes if he does 1 to 10 chores in a week, and then represent this function in a table of values, a set of ordered pairs, and a graph.Use function notation to represent the relation between Fahrenheit and Celsius.
You want to make tanghulu, a treat made of fruit in a candy coating. To make the candy, you need to heat the sugar to 149°C, but your thermometer is only marked in Fahrenheit. Fortunately, the function f(x)=1.8x+32 converts temperature in Celsius to temperature in Fahrenheit. The function is written in function notation, and f(x) is read as “f of x”, meaning a function of x. Function notation is helpful because it allows you to see the ordered pair directly from the function notation. For example, to calculate the temperature you need, you first substitute 149°C everywhere there is an “x”, giving f(149)=1.8(149)+32. Then evaluate: f(149)=300.2. In the function notation you can directly see the xvalue that produces each output or yvalue: f(x)=y. You can easily write the ordered pair (149, 300.2). Now you know that you need to heat the sugar to 300.2°F when your make your tanghulu. Try this yourself: A fudge recipe requires you to cook the sugar to the soft ball stage of 113°C, what is this temperature in degrees Fahrenheit?
Use function notation to represent the relation between Fahrenheit and Celsius. You want to make tanghulu, a treat made of fruit in a candy coating. To make the candy, you need to heat the sugar to 149°C, but your thermometer is only marked in Fahrenheit. Fortunately, the function f(x)=1.8x+32 converts temperature in Celsius to temperature in Fahrenheit. The function is written in function notation, and f(x) is read as “f of x”, meaning a function of x. Function notation is helpful because it allows you to see the ordered pair directly from the function notation. For example, to calculate the temperature you need, you first substitute 149°C everywhere there is an “x”, giving f(149)=1.8(149)+32. Then evaluate: f(149)=300.2. In the function notation you can directly see the xvalue that produces each output or yvalue: f(x)=y. You can easily write the ordered pair (149, 300.2). Now you know that you need to heat the sugar to 300.2°F when your make your tanghulu. Try this yourself: A fudge recipe requires you to cook the sugar to the soft ball stage of 113°C, what is this temperature in degrees Fahrenheit? 
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When exchanging money, each US dollar is the same as 1.2 Canadian dollars. Is the relationship between US dollars and Canadian dollars a function?
Is the following set of ordered pairs a function? Why or why not? {(2, 3), (2, 4), (3, 7), (4, 12)}
Convert the following equation to function notation and then find the value of the function when x is equal to 4. y=3x6
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