1 Chapter 2 Section 1 INPUT AND

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Chapter 2 Section 1 INPUT AND OUTPUT 2

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

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) ? Page 62 4

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

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

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

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

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) Page 62 9

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

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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? Page 62 Example 2 16

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

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

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

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

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

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 2 x (factor of 2), we increase the Area by 4 x (factor of 4). Or, we double r we quadruple A. Page N/A 22

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

g(3): Page 62 24

g(-1): Page 62 25

g(a): Page 62 26

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

h(2): Page 63 28

h(2): Page 63 29

h(a-2): Page 63 30

h(a-2): Page 63 31

h(a)-2: Page 63 32

h(a)-2: Page 63 33

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

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

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) Page 63 36

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

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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? Page 64 Example 7 39

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

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

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 1975. (a) Evaluate and interpret f(25). (b) Solve and interpret f(t) = 1159. Page 64 Example 8 43

R = f(t) (a) Evaluate and interpret f(25). (b) Solve and interpret f(t) = 1159. Year, t (since 1975) 0 5 10 15 20 25 30 Revenue, R 201 364 651 1075 1159 2200 (million $) Page 64 44

R = f(t) (a) Evaluate and interpret f(25). f(25) = 2200. Therefore, in 2000 (1975+25), revenue was $2, 200 million. Year, t (since 1975) 0 5 10 15 20 25 30 Revenue, R 201 364 651 1075 1159 2200 (million $) Page 65 45

R = f(t) (b) Solve and interpret f(t) = 1159. Year, t (since 1975) 0 5 10 15 20 25 30 Revenue, R 201 364 651 1075 1159 2200 (million $) Page 65 46

R = f(t) (b) Solve and interpret f(t) = 1159. When were Revenues $1159 million? Year, t (since 1975) 0 5 10 15 20 25 30 Revenue, R 201 364 651 1075 1159 2200 (million $) Page 65 47

R = f(t) When were Revenues $1159 million? t=20. Therefore, 1995. Year, t (since 1975) 0 5 10 15 20 25 30 Revenue, R 201 364 651 1075 1159 2200 (million $) Page 65 48

End of Section 2. 1 49