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4.1 Notes.notebook
1
February 15, 2013
Mar 1512:48 PM
Unit 4:Combining Functions
4.1 Combining Functions GraphicallyFunctions can be added, subtracted, multiplied or divided withother functions. The result will be a new function in most cases.
We will look at this from both a graphical and algebraic view.
4.1 Notes.notebook
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February 15, 2013
Mar 1512:54 PM
Let us begin by looking at the following functions:
f(x) = 2x + 6 g(x) = x + 1
Place both of these functions into your calculator:
We will add the functions together to produce a new function h(x).
h(x)=f(x) + g(x). This is often called (f+g)(x).
Look at the viewscreen to set this up on your calculator:
Notice that Y3 is showing thesum of Y1 and Y2 and is thereforeequivalent to h(x).
4.1 Notes.notebook
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February 15, 2013
Mar 151:06 PM
When we observe the graph of this relationship let us look forpatterns:
Y1 Y2Y3 :h(x) the sum of Y1 and Y2
Input various x values into the calculator and notice thatY3 is always the sum of Y1 and Y2(use your UP arrow key to jump from one function to the next)
Also notice that the domain of h(x) is "all reals" since both f(x) and g(x) have that domain.
4.1 Notes.notebook
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February 15, 2013
Mar 151:13 PM
Now change Y3 to be Y1Y2.We have now change h(x) to be f(x)g(x).While the order for division does not matter we know that it DOES make a difference for subtraction.
Now we are looking at (fg)(x).
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February 15, 2013
Mar 151:16 PM
Here is what your graph should look like:
Y1
Y 2
Y3
Notice that the Y3 values for various x values,are the results you get when you go Y1 Y2.REVERSE the order of the subtraction and noticea new equation is created in Y3.
4.1 Notes.notebook
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February 15, 2013
Mar 151:22 PM
Change the operation in Y3 to multiplication andthen your graph screen will look like this:
Y1
Y2
Y3
Again, check for various x inputs, and the resultsalways show that Y3 = Y1*Y2.It is also interesting that the resulting product EQUATIONis NOT linear. Think about the equation you would getwhen you multiply (2x+6) by (x+1) and youshould understand the QUADRATIC result.
4.1 Notes.notebook
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February 15, 2013
Mar 151:28 PM
Our final operation is division. The resulting screen is very interesting.
Y1
Y2
Y3
When we input various x values the result for Y3 is alwaysY1/Y2. Why therefore is there a vertical asymptote at x=1.Division requires us to worry about the equation in thedenominator and remember this important fact:
Even if both functions have Domains of ALL REALS,the function that results from the operation of division, has a domain that is RESTRICTED by the NONPERMISSIBLEvalues of the denominator function. In this example:
is
and
4.1 Notes.notebook
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February 15, 2013
Mar 151:41 PM
Look at the following graph from PAGE 270 #2.Let Y1 be f(x)=Let Y2 be g(x) = 2x+4The graph on your calculator should look like:
By using various points PLOT a sketch of f(x)*g(x).Notice how x=2 and x=2 are obviouspoints to use. Verify your sketch by creating theappropriate Y3 equation.
Because the domain of f(x) is x>2,what has happened to the domain of (f*g)(x)??
4.1 Notes.notebook
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February 15, 2013
Mar 151:44 PM
Look at pg. 271 #3. Do the question without your calculator.
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February 15, 2013
Mar 151:50 PM
Conclusions:
1) Domain of function will be the same as domain ofmost restricted function for adding, subtracting andmultiplication operations.
2) Range is more difficult to determine and if requestedis usually a calculator question.
3) Domain if operation is division is complicated by non permissible values in denominator.
Verify these conclusions by placing the following in yourcalculator and attempting all 4 operations:
4.1 Notes.notebook
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February 15, 2013
Mar 1611:53 AM
HOMEWORK:Supplementary Sheet 1: #120
4.1 Notes.notebook
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February 15, 2013
Mar 1610:50 AM
4.2 Combining Functions Algebraically
Now that we have seen the graphs of combined functions,and realize that when we add 2 functions the result is the sumof their y values, we can realize that this result is the sum asALGEBRAICALLY combining their equations to produce a newequation.This new equation will automatically give the same value as thesum of the 2 original equations. This process also works for the other 3 operations.
Ex) Put f(x) = 3x8 and g(x) = 2x+5 into your calculator.Graph them and also graph (f+g)(x).Test the values you get for x=1f(1)= ______ g(1) = _______ (f+g)(1) = ______Now ALGEBRAICALLY add f(x) to g(x) and call it h(x)
h(x) =____________________________
Now find h(1)=_______________
4.1 Notes.notebook
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February 15, 2013
Mar 1611:01 AM
Example)
Note: We would have to graph d(x) and p(x) to determine the range of eachwith a calculator.
d) Find f(4), g(4), d(4), and p(4) to verify that the combined functionsdo produce the correct values for the operations used.
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February 15, 2013
Feb 153:19 PM