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INPUT AND OUTPUT Chapter 2 Section 1 2

You will remember the following problem from Chapter 1, Section 1: 3Page 62

The number of gallons of paint needed to paint a house depends on the size of the house. A gallon of paint typically covers 250 square feet. Thus, the number of gallons of paint, n, is a function of the area to be painted, A ft 2. We write n = f(A). Which is the output and which is the input for: n = f(A) ? 4Page 62

A reminder from Chapter 1: Output = f(Input) Or: Dependent = f(Independent) 5Page 4

Which is the output and which is the input for: n = f(A) ? 6Page 62

Which is the output and which is the input for: n = f(A) ? n=f(A) => output, A => input 7Page 62

n=f(A) => output, A => input For example, f(20,000) represents ? 8Page 62

n=f(A) => output, A => input f(20,000) represents the # of gallons of paint to cover a house of 20,000 sq ft. (ft 2 ) 9Page 62

Using the fact that 1 gallon of paint covers 250 ft 2, evaluate the expression f(20,000). 10Page 62 Example 1

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Area of a circle of radius r: A = q(r) = πr 2. Use the formula to evaluate q(10) and q(20). What do your results tell you about circles? 16Page 62 Example 2

Area of a circle of radius r: A = q(r) = πr 2. Use the formula to evaluate q(10) and q(20). 17Page 62

Area of a circle of radius r: A = q(r) = πr 2. Use the formula to evaluate q(10) and q(20). 18Page 62

Area of a circle of radius r: A = q(r) = πr 2. Use the formula to evaluate q(10) and q(20). 19Page 62

Area of a circle of radius r: A = q(r) = πr 2. Use the formula to evaluate q(10) and q(20). 20Page 62

Area of a circle of radius r: A = q(r) = πr 2. What do your results tell you about circles? 21Page N/A

Area of a circle of radius r: A = q(r) = πr 2. What do your results tell you about circles? If we increase the radius by 2x (factor of 2), we increase the Area by 4x (factor of 4). Or, we double r  we quadruple A. 22Page N/A

Let: Evaluate: g(3), g(-1), g(a) 23Page 62 Example 3

g(3): 24Page 62

g(-1): 25Page 62

g(a): 26Page 62

Let h(x) = x 2 − 3x + 5. Evaluate and simplify the following expressions. (a) h(2) (b) h(a − 2) (c) h(a) − 2 (d) h(a) − h(2) 27Page 63 Example 4

h(2): 28Page 63

h(2): 29Page 63

h(a-2): 30Page 63

h(a-2): 31Page 63

h(a)-2: 32Page 63

h(a)-2: 33Page 63

h(a)-h(2): 34Page 63

h(a)-h(2): 35Page 63

Finding Input Values: Solving Equations Given an input, we evaluate the function to find the output. (Input  Output) Sometimes the situation is reversed; we know the output and we want to find the corresponding input. (Output  Input) 36Page 63

Back to the "Cricket" function, but now if T = 76, R = ? 37Page 63 Example 5

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Area of a circle of radius r (cm.): A = q(r) = πr 2. What is the radius of a circle whose area is 100 cm 2 ? 39Page64 Example 7

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Since a circle CAN'T have a negative radius, we conclude: 41Page 64

Finding Output and Input Values from Tables and Graphs 42Page 64

Table 2.1 shows the revenue, R = f(t), received or expected, by the National Football League, 1 NFL, from network TV as a function of the year, t, since (a) Evaluate and interpret f(25). (b) Solve and interpret f(t) = Page 64 Example 8

R = f(t) (a) Evaluate and interpret f(25). (b) Solve and interpret f(t) = Year, t (since 1975) Revenue, R (million $) Page 64

R = f(t) (a) Evaluate and interpret f(25). f(25) = Therefore, in 2000 ( ), revenue was $2,200 million. Year, t (since 1975) Revenue, R (million $) Page 65

R = f(t) (b) Solve and interpret f(t) = Year, t (since 1975) Revenue, R (million $) Page 65

R = f(t) (b) Solve and interpret f(t) = When were Revenues $1159 million? Year, t (since 1975) Revenue, R (million $) Page 65

R = f(t) When were Revenues $1159 million? t=20. Therefore, Year, t (since 1975) Revenue, R (million $) Page 65

End of Section