Understanding Functions An Inter-Dimensional Junk Food Adventure.
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Transcript of Understanding Functions An Inter-Dimensional Junk Food Adventure.
![Page 1: Understanding Functions An Inter-Dimensional Junk Food Adventure.](https://reader036.fdocuments.net/reader036/viewer/2022062718/56649e8f5503460f94b927ea/html5/thumbnails/1.jpg)
Understanding Functions
An Inter-DimensionalJunk Food Adventure
![Page 2: Understanding Functions An Inter-Dimensional Junk Food Adventure.](https://reader036.fdocuments.net/reader036/viewer/2022062718/56649e8f5503460f94b927ea/html5/thumbnails/2.jpg)
This is me,
And this is a function
![Page 3: Understanding Functions An Inter-Dimensional Junk Food Adventure.](https://reader036.fdocuments.net/reader036/viewer/2022062718/56649e8f5503460f94b927ea/html5/thumbnails/3.jpg)
And this is an Inter-Dimensional Vending Machine
It will all make sense before the end, I promise.
And now a question: What aremathematical functions?
![Page 4: Understanding Functions An Inter-Dimensional Junk Food Adventure.](https://reader036.fdocuments.net/reader036/viewer/2022062718/56649e8f5503460f94b927ea/html5/thumbnails/4.jpg)
The Gist of it: functions express relationships
Lets say we have an equation, like this one:
This equation expresses a relationship between x and y
It says that if we are given the value of x, then we can determine the value of y. For example, if x = 2, then y = 9.
Being able to determine y using only x is what it means to say that y is a function of x.
![Page 5: Understanding Functions An Inter-Dimensional Junk Food Adventure.](https://reader036.fdocuments.net/reader036/viewer/2022062718/56649e8f5503460f94b927ea/html5/thumbnails/5.jpg)
Writing it down: function notation
Recalling our previous equation
If we want to express that y is a function of x then we should use function notation.
we can write y = f(x) to mean "y is a function of x". We often cut out the middle man, y, and just write this
![Page 6: Understanding Functions An Inter-Dimensional Junk Food Adventure.](https://reader036.fdocuments.net/reader036/viewer/2022062718/56649e8f5503460f94b927ea/html5/thumbnails/6.jpg)
But there's more: functions are predictableWe can think of a function as a kind of machine that produces an output when it is given an input.
Why not? Well, if x is anything besides zero, then y is ambiguous. E.g. inputing x=3, outputs either 3 or -3 for y.
For example, this is an expression in which y is not a function of x :
This machine is predictable, and does not give two different outputs for the same input. Instead, one input leads to one output. This is also true of functions.
input output
![Page 7: Understanding Functions An Inter-Dimensional Junk Food Adventure.](https://reader036.fdocuments.net/reader036/viewer/2022062718/56649e8f5503460f94b927ea/html5/thumbnails/7.jpg)
Some Terminology: domain and range
We have been using the terms input and output instead of the usual terms domain and range.
The domain of a function f(x) is anything we can put in for x, and the range of the function is the collection of all of the outputs.
![Page 8: Understanding Functions An Inter-Dimensional Junk Food Adventure.](https://reader036.fdocuments.net/reader036/viewer/2022062718/56649e8f5503460f94b927ea/html5/thumbnails/8.jpg)
What about that vending machine?
Suppose that we found a vending machine. Not just any vending machine, but a vending machine that is situated in a multi-dimensional nexus. It never runs out of snacks.
As it turns out, we can think of this inter-dimensional marvel as a function.
![Page 9: Understanding Functions An Inter-Dimensional Junk Food Adventure.](https://reader036.fdocuments.net/reader036/viewer/2022062718/56649e8f5503460f94b927ea/html5/thumbnails/9.jpg)
What is its domain?
Like most vending machines, you input button presses for the snack you'd like.
If the vending machine is a function, then ...
What is its range?
When you input some button presses, you get a snack in return.
Is it predictable?
It had better be! The button for a Choco-Explosion should spit out a Choco-Explosion every time, and nothing else.
![Page 10: Understanding Functions An Inter-Dimensional Junk Food Adventure.](https://reader036.fdocuments.net/reader036/viewer/2022062718/56649e8f5503460f94b927ea/html5/thumbnails/10.jpg)
For Example
The domain is the set of buttons
The range is the set of snacks
pressing buttons corresponds to plugging in values for x.
![Page 11: Understanding Functions An Inter-Dimensional Junk Food Adventure.](https://reader036.fdocuments.net/reader036/viewer/2022062718/56649e8f5503460f94b927ea/html5/thumbnails/11.jpg)
A broken machine: recognizing a non-function
Suppose that there is a glitch in the machine.
We press C3 and hope to get a Mega-Choco Wedge like before. But this time, a Plutonian Sugar Spin comes out instead.
This is a clear sign that the machine is dysfunctional, because outputs should be determined by inputs.
![Page 12: Understanding Functions An Inter-Dimensional Junk Food Adventure.](https://reader036.fdocuments.net/reader036/viewer/2022062718/56649e8f5503460f94b927ea/html5/thumbnails/12.jpg)
Other representations: the many faces of functions
Here is a function in function notation:
And here is a table containing some values of the function for different values of x:
And here is a graph for this function, where y=f(x)
x f(x)1 4-2 133 18
![Page 13: Understanding Functions An Inter-Dimensional Junk Food Adventure.](https://reader036.fdocuments.net/reader036/viewer/2022062718/56649e8f5503460f94b927ea/html5/thumbnails/13.jpg)
One more thing: evaluating a function
Given a function such as
Then we can evaluate the function by "plugging in" values for x. For example:
So, evaluating f(x) for x=7, gives 22.
![Page 14: Understanding Functions An Inter-Dimensional Junk Food Adventure.](https://reader036.fdocuments.net/reader036/viewer/2022062718/56649e8f5503460f94b927ea/html5/thumbnails/14.jpg)
Wrapping Up
Key points:
• y = f(x) says that the y is a function of x - when we are given x, we can find y.
• The written form f(x) is called function notation.• A function is predictable - two different outputs cannot
be associated with any single input.• Functions can be evaluated by plugging in values for x. • Functions have domain and range, which loosely
correspond to sets of input and output.
![Page 15: Understanding Functions An Inter-Dimensional Junk Food Adventure.](https://reader036.fdocuments.net/reader036/viewer/2022062718/56649e8f5503460f94b927ea/html5/thumbnails/15.jpg)