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Main ideas in this unit

In this unit, students learn about functions, building on their work in middle school. A function is a relationship between an input and an output, where for every input there is exactly one output. Here are some examples of functions:

We often use the phrase “(output) is a function of (input)” to express how the input and output sets are related. For example, “the number of letters in a name is a function of the name,” or “the temperature in the oven is a function of time since it was turned on.”

To make it easier to talk about and work with functions, we often use letters to name them, and we use function notation to represent their input and output.

Suppose is a function that tells us the distance, in feet, that a child ran over time, tt, in seconds. So: ff is the name of the function, time is the input, and distance is the output.

Here is how we represent this information in function notation:

The notation is read “ff of tt”.

Here are examples of some things we can say with function notation:

in tt seconds, the child ran 50 feet

A function can also be represented with a graph. Here is a graph of function ff.

We can use it to estimate the input and output values of the function.

For instance, the graph shows that f(3)=10f(3)=10, which means that 3 seconds after she started running, the child has run 10 feet.

We can also use the graph to find out the time when the child has run 50 feet, or the value of tt in f(t)=50f(t)=50. We can see that it happened when tt is 18.

Sometimes a rule tells us what to do to the input of a function to get the output.

Suppose function gives the dollar cost of buying burritos at $5 each. To get the output (the cost), we multiply the input (the number of burritos) by 5. We can write: g(x)=5xg(x)=5x.

The height of a plant in centimeters is a function of its height in inches, .

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