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BY DR. JULIA ARNOLD The Algebra of Functions

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What does it mean to add two functions? If f(x) = 2x + 3 and g(x) = -4x - 2 What would (f+g)(x) be? (f+g)(x) = f(x) + g(x) It means to add the two functions (f+g)(x) = -2x + 1 Likewise (f - g)(x) = f(x) - g(x) or (f - g)(x) = 6x +5 Multiplication of two functions is expressed like this: fg(x) = f(x)g(x) In our example, fg(x) = -8x 2 -16x -6

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The Algebra of Functions (f+g)(x) = f(x) + g(x) (f - g)(x) = f(x) - g(x) fg(x) = f(x)g(x) Division also follows a logical path: In our example: f(x) = 2x + 3 and g(x) = -4x - 2

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Application: In business you would have fixed costs, such as rent, and variable costs from producing your commodity. We will call the cost C(x) Revenue is the money you make in your business. We will call revenue R(x). Profit is what you hope to make from your business and is denoted as P(x) = R(x) - C(x).

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Suppose your company manufactures water filters and has fixed costs of $10,000 per month. The cost of producing the water filters is represented by -.0001x 2 + 10x where How do you represent the cost function? C(x) = -.0001x 2 + 10x +10000 Suppose the total revenue you make from the sale of x water filters is given by R(x) = -.0005x 2 + 20x What would the profit function be? How much would you make if you sold 10,000 water filters?

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Suppose your company manufactures water filters and has fixed costs of $10,000 per month. The cost of producing the water filters is represented by -.0001x 2 + 10x where How do you represent the cost function? C(x) = -.0001x 2 + 10x +10000 Suppose the total revenue you make from the sale of x water filters is given by R(x) = -.0005x 2 + 20x What would the profit function be? P(x) = R(x) - C(x) = -.0005x 2 + 20x - (-.0001x 2 + 10x +10000) P(x)= (-.0005 +.0001) x 2 +20x -10x - 10000 P(x) = -.0004x 2 +10x -10000 How much would you make if you sold 10,000 water filters? P(10000)= -.0004(10000) 2 +10(10000) - 10000 = 50,000 per month

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Composition of Functions (one more operation) The easiest way to describe composition is to say it is like substitution. In fact Read f of g of x which means substitute g(x) for x in the f(x) expression. Suppose f(x)= 2x + 3 and g(x) = 8 - x f(g(x) )= 2 g(x) + 3 f(8 - x)= 2 (8 - x) + 3 f(g(x)) = 16 -2x + 3 or 19 - 2x

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An interesting fact is that most of the time. Let’s see if this is the case for the previous example.

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f(x) = 2x + 3, andg(x) = 8 - x Thus we will substitute f into g. g(x) = 8 - x g(f(x) ) = 8 - f(x) Now substitute what f(x) is: g(2x + 3) = 8 - (2x + 3) = 8 - 2x - 3 = 5 - 2x f(g(x)) = 19 - 2x while g(f(x)= 5 - 2x

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Write the f function Substitute g(x) for x Replace g(x) with Simplify Step 1 Step 2 Step 3 Step 4

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Find: When ready click your mouse. The answer is: A) B) Move your mouse over the correct answer.

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Find: When ready click your mouse. The answer is: B) A) Move your mouse over the correct answer.

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Find: When ready click your mouse. The answer is: B) A) Move your mouse over the correct answer.

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We can also evaluate the composition of functions at a number. Let: and Find

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= f(g(3)) This says to insert the value for g(3) into f, so… Step 1 is to find g(3)

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Find = f(g(3)) Now substitute the answer into f(x) for x. 12 34 5 1: Take the square root of top and bottom. 2: Find a number that rationalizes the denominator 3: multiply top and bottom4: Take the square root of 36 5: add the 1 as 6/6

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