Algebra of Functions Let f and g be functions with domains D and E respectively. Define new functions from D E -> R by (f+g)(x) = f(x) + g(x) (f-g)(x) = f(x) - g(x) (f-g)(x) = f(x) - g(x) (fg)(x) = f(x) * g(x) (fg)(x) = f(x) * g(x) (f/g)(x) = f(x) / g(x) if g (x) 0 (f/g)(x) = f(x) / g(x) if g (x) 0

f + g (f+g)(x) = f(x) + g(x) f(-0.8) = 0.7 g(-0.8) = -0.7 (f+g)(-0.8) = 0

f + g (f+g)(x) = f(x) + g(x) (f+g)(x) = (3 – x) + [4 – (x – 1) 2 ] (f+g)(x) = (3 – x) + [4 – (x – 1) 2 ] (f+g)(1) = (3-1) + [4 – 0 2 ] = = 6

f - g (f - g)(x) = f(x) - g(x) = (3 – x) – [4 – (x – 1) 2 ] = -1 – x + [x 2 -2x +1] = x 2 – 3x = x(x – 3) (f – g)(1) = 1(1 – 3) = -2 or (f – g)(1) = 2 – 4 = -2

g – f (g - f)(x) = g(x) - f(x) = [4 – (x – 1) 2 ] - (3 – x) = 4 - [x 2 - 2x +1] – 3 + x = 1 - x 2 + 2x -1 + x = - x 2 + 3x = -x(x – 3) (g - f)(1) = 4 – 2 = 2

Find f + g A. (f+g)(x)=7 – x – (x – 1) 2 B. (f+g)(x)= x(x - 3) C. (f+g)(x)= -x(x - 3)

Evaluate (f+g)(0)

f*g(x) = f(x) g(x) If f(x) = x 2 -1 If f(x) = x 2 -1 and g(x) = x find (fg)(2) (f*g)(2) = f(2)g(2) = 3 (14) = 42

f/g(x) = f(x) / g(x) If f(x) = x 2 -1 If f(x) = x 2 -1 and g(x) = x find (f/g)(2) (f/g)(2) = f(2) / g(2) = 3 / 14 = 0.214

If f(x) = x 2 -1 and g(x) = x find (f*g)(1) If f(x) = x 2 -1 and g(x) = x find (f*g)(1)

If f(x) = x 2 -1 and g(x) = x find (f/g)(2) If f(x) = x 2 -1 and g(x) = x find (f/g)(2)

10/6/ Tan Applied Calculus 2.3 f(0) = Domain of f = Range of f = Find x so f(x) = 3.

10/6/ Tan Applied Calculus 2.3 Manufacture of speakers has monthly fixed costs of $23,000 and a production cost of $15.00 for each speaker. What is the cost function? C(x) = 15 x worker materialtransportation

10/6/ C(x) = 15 x What is the slope? 15 What is the cost of producing speakers? What should we sell them for to break even? 16.15

10/6/ C(x) = 15 x If the domain is [0, 40000], what is the range? [0, ] If we sell them for 16.15, find the revenue function.

10/6/ R(x) = x If we sell them for 16.15, find the revenue function. What is the slope? What is the revenue if we sell 40000?

Profit is P(x) = R(x) – C(x) R(x) = x C(x) = 15 x R(x) = x C(x) = 15 x P(x) = x – (15 x ) P(x) = x – (15 x ) P(x) = 1.15 x P(x) = 1.15 x P(40000) = P(40000) = /6/ Money out Money in

Find the total cost function. A monthly fixed cost of $10000 A monthly fixed cost of $10000 and a variable cost of x x on the domain [0, 40000] C(x) = x x C(x) = x x

Find the total cost function. A monthly fixed cost of $10000 A monthly fixed cost of $10000 f(x) = g(x) = x x

Domain = [0, 40000] Domain = [0, 40000] Range = Range = Fixed cost Fixed cost

C(x) = x x What is the total cost for producing each month? C(30000) = (30000) = $220000

Find the total cost of producing 1000 each month C(x) = x x C(1000) = ( ) = = $19900

C(x) = x x Find the cost for making

The Revenue function. The monthly revenue function R(x) = The monthly revenue function R(x) = x x on [0, 40000] How much revenue is received for the sale of 1000 in a month? How much revenue is received for the sale of 1000 in a month? R(1000) = R(1000) = ( ) = $19500

R(x) = x x How much received for 100?

Find the profit function. A revenue function of R(x) = A revenue function of R(x) = x x on [0, 40000] C(x) = x x C(x) = x x

P(x) = R(x) – C(x) A revenue function of R(x) = A revenue function of R(x) = x x on [0, 40000] C(x) = x x C(x) = x x P(x) = P(x) = x x – [ x x ] = x x

P(x)= x x Find the profit if the company produces a month. P(10000) = (10000*10000) – = $50000 a month.

(f/g)(x) = f(x) / g(x)

f*g(x) = f(x) g(x)

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Algebra of Functions Let f and g be functions with domains D and E respectively. Define new functions from D’ -> R by (g o f)(x) = g(f(x)) D’ = {x|f(x)  E}

Algebra of Functions Let f and g be functions with domains D and E respectively. Define new functions from E’ -> R by (f o g)(x) = f(g(x)) E’ = {x|g(x)  D}

Composition of functions f(x) = x 2 g(x) = Evaluate f o g(0) and g o f (0)

g(x) = x 2 and f(x) = evaluate g o f(-2)

g(x) = x 2 and f(x) = evaluate f o g(-2)