• Ex 4:  Domain of a Composite Function
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    Ex 4: Domain of a Composite Function

    – GIVEN F OF X AND G OF X, WE WANT TO FIND THE COMPOSITE FUNCTION, F OF G, AND THEN DETERMINE THE DOMAIN OF F OF G. WHEN DETERMINING THE DOMAIN OF COMPOSITE FUNCTIONS, WE DO NEED TO BE VERY CAREFUL. THE DOMAIN OF THE COMPOSITE FUNCTION F OF G OF X MUST CONTAIN THE RESTRICTIONS OF THE DOMAIN OF THE INNER FUNCTION, G OF X, AS WELL AS THE RESTRICTIONS ON THE NEWLY FORMED COMPOSITE FUNCTION. SO, AGAIN, WE NEED TO CONSIDER THE DOMAIN OF THE INNER FUNCTION, G OF X, AS WELL AS THE DOMAIN OF THE COMPOSITE FUNCTION TO DETERMINE THE DOMAIN OF F OF G…

  • Ex 2:  Determine the Domain and Range of the Graph of a Function
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    Ex 2: Determine the Domain and Range of the Graph of a Function

    – WE WANT TO DETERMINE THE DOMAIN AND RANGE OF A FUNCTION GIVEN THE GRAPH OF THE FUNCTION. THE DOMAIN IS A SET OF ALL POSSIBLE X VALUES OF THE FUNCTION. X VALUES OCCUR ALONG THE X AXIS OR THE HORIZONTAL AXIS. AND THE RANGE IS A SET OF ALL POSSIBLE Y VALUES OF THE FUNCTION, AND Y VALUES OCCUR ALONG THE VERTICAL AXIS. SO IF WE’RE GIVEN THE GRAPH OF A FUNCTION, AND WE WANT TO DETERMINE THE DOMAIN OF THE FUNCTION, WE WANT TO PROJECT THE GRAPH ONTO THE X AXIS, OR DETERMINE HOW THE GRAPH BEHAVES HORIZONTALLY ALONG THE X AXIS. WHAT I MEANT BY THAT IS…

  • Ex 3:  Domain of a Composite Function
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    Ex 3: Domain of a Composite Function

    – GIVEN F OF X AND G OF X, WE WANT TO FIND THE COMPOSITE FUNCTION F OF G, AND THEN DETERMINE THE DOMAIN OF THE COMPOSITE FUNCTION F OF G. WE NEED TO BE CAREFUL WHEN DETERMINING THE DOMAIN OF A COMPOSITE FUNCTION BECAUSE THE DOMAIN OF F OF G OF X MUST CONTAIN THE RESTRICTIONS OF THE DOMAIN OF THE INNER FUNCTION G OF X, AS WELL AS THE RESTRICTIONS ON NEWLY FORM COMPOSITE FUNCTION. SO TO START, IT’S GOING TO BE HELPFUL TO WRITE OUR COMPOSITE FUNCTION USING THIS DEFINITION HERE. SO WE CAN WRITE F OF G, SOMETIMES WRITTEN LIKE THIS OR EVEN LIKE THIS CAN…

  • Ex:  Determine the Domain of a Square Root Function
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    Ex: Determine the Domain of a Square Root Function

    – WE WANT TO DETERMINE THE DOMAIN AND RANGE OF THE FUNCTION F(X)=SQUARE ROOT OF 2X – 4. THERE’S A COUPLE WAYS OF DOING THIS BASED UPON HOW MUCH WE KNOW ABOUT A GIVEN FUNCTION. IF YOU’RE MORE OF A VISUAL PERSON, THE BEST THING TO DO IS GRAPH THE FUNCTION AND THEN DETERMINE THE DOMAIN AND RANGE BY ANALYZING THE GRAPH. LET’S START BY DOING THAT AND THEN WE’LL USE A SECOND METHOD TO DETERMINE THE DOMAIN AND RANGE. SO WHETHER YOU MAKE A GRAPH USING TECHNOLOGY OR THE TABLE OF VALUES, THE GRAPH WOULD LOOK LIKE THIS. REMEMBER THE DOMAIN IS A SET OF ALL POSSIBLE X VALUES…

  • Physical Computing: Switching LEDs with an Arduino or Raspberry Pi
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    Physical Computing: Switching LEDs with an Arduino or Raspberry Pi

    In this video about “Physical Computing” I will show you, how to switch LEDs using GPIOs I am using a Raspberry Pi 3 and an Arduino Uno for the demonstrations. GPIO means General Purpose Input Output, thus you can use those pins in either input or output mode by changing nothing but a few lines of software code. In order to switch a load as demonstrated in this video, the output mode is needed. In output mode we can read either 0V… …or 3.3V at the GPIO of the Raspberry Pi… …and 5V at the pin of the Arduino. That voltage is recorded between ground and the GPIO pin. If…

  • Ex:  Find the Inverse of a Square Root Function with Domain and Range
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    Ex: Find the Inverse of a Square Root Function with Domain and Range

    – WE’RE GIVEN F OF X EQUALS THE SQUARE ROOT OF THE QUANTITY (2X – 1) – 3. WE WANT TO DETERMINE THE DOMAIN AND RANGE OF THE GIVEN FUNCTION AND THEN FIND THE INVERSE FUNCTION. BECAUSE OUR FUNCTION CONTAINS A SQUARE ROOT IN ORDER FOR THE FUNCTION VALUE TO BE REAL THE NUMBER UNDERNEATH THE SQUARE ROOT OR THE RADICAND WHICH IN THIS CASE 2X – 1 CAN’T BE NEGATIVE WHICH MEANS 2X – 1 MUST BE GREATER THAN OR EQUAL TO ZERO. SINCE 2X – 1 MUST BE GREATER THAN OR EQUAL TO ZERO THIS RESTRICTION WILL HELP US FIND OUR DOMAIN. WE JUST NEED TO SOLVE THIS…

  • Physical computing: HowTo control LEDs, electric motors or servos through GPIOs
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    Physical computing: HowTo control LEDs, electric motors or servos through GPIOs

    In this video I would like to explain how to amplify digital signals, so that you can control homebuilt peripherals by computers or microcontrollers. Furthermore simple input functionalities are treated. The control units I am using are an Arduino Uno microcontroller and a Raspberry Pi single board computer. Both computing machines have digital input/output pins that can be used to control peripherals. Before connecting a device to a pin you must consider the electrical properties of the ports! The Arduino Uno operates with a pin voltage of 5V, while the voltage at the pins of the Raspberry Pi must be kept below 3.3V. First let’s have a look at the…

  • Ex:  Determine the Domain of an Absolute Value Function
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    Ex: Determine the Domain of an Absolute Value Function

    – WE WANT TO DETERMINE THE DOMAIN AND RANGE OF THE FUNCTION F(X)= THE ABSOLUTE VALUE OF X + 2 – 3. THE DOMAIN CONSISTS OF ALL POSSIBLE INPUTS OR X VALUES FOR THE FUNCTION AND THE RANGE CONSISTS OF ALL POSSIBLE OUTPUTS OR Y VALUES FOR THE FUNCTION. NOW THERE ARE A COUPLE OF WAYS OF DETERMINING THE DOMAIN AND RANGE. WE COULD USE WHAT WE KNOW ABOUT THE PROPERTIES OF THE ABSOLUTE VALUE FUNCTION TO DETERMINE THE DOMAIN AND RANGE OR WE COULD ALSO MAKE A GRAPH OF THE FUNCTION AND ANALYZE THE GRAPH. LET’S START BY ANALYZING THE GRAPH. SO THE FIRST STEP WOULD BE TO EITHER…

  • Ex: Give the Domain and Range Given the Points in a Table
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    Ex: Give the Domain and Range Given the Points in a Table

    – THE FUNCTION F OF X IS DEFINED IN THE TABLE BELOW. USING THE TABLE IDENTIFY THE DOMAIN AND THE RANGE OF THE FUNCTION. WE’RE ASKED TO WRITE THE ANSWERS AS A ORDERED LIST ENCLOSED IN CURLY BRACKETS. FOR REVIEW, THE DOMAIN IS A SET OF ALL POSSIBLE INPUTS, OR IN THIS CASE, THE X VALUES. AND THE RANGE IS A SET OF ALL POSSIBLE OUTPUTS OR Y VALUES, WHICH WOULD BE THE FUNCTION VALUES. AND THEREFORE, THIS COLUMN REPRESENTS THE POSSIBLE INPUTS OR THE DOMAIN, AND THIS COLUMN REPRESENTS THE POSSIBLE OUTPUTS OR FUNCTION VALUES, WHICH WOULD BE THE RANGE. SO WE’RE ASKED TO INCLUDE THE DOMAIN AND RANGE…

  • Ex 1:  Determine the Domain and Range of the Graph of a Function
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    Ex 1: Determine the Domain and Range of the Graph of a Function

    – WE WANT TO DETERMINE THE DOMAIN AND RANGE OF A FUNCTION GIVEN THE GRAPH OF THE FUNCTION. A DOMAIN IS A SET OF ALL POSSIBLE X VALUES FOR THE FUNCTION, AND THE RANGE IS A SET OF ALL POSSIBLE Y VALUES FOR THE FUNCTION. WELL, X VALUES ARE ALONG THE HORIZONTAL AXIS, AND Y VALUES ARE ALONG THE VERTICAL AXIS. SO TO HELP US DETERMINE THE DOMAIN OF THIS FUNCTION, WE WANT TO PROJECT THIS FUNCTION ONTO THE X AXIS OR ANALYZE IT TO DETERMINE HOW THE GRAPH BEHAVES FROM LEFT TO RIGHT. SO IF WE WERE TO PROJECT THIS FUNCTION ON TO THE X AXIS, OR IF WE…