2.1 INPUT AND OUTPUT 1 Finding Output Values: Evaluating a Function Example 4 Let h(x) = x2 − 3x + 5. Evaluate and simplify the following expressions. (a) h(2) (b) h(-3) (c) h(a) (d) h(a) − h(2) Solution Notice that x is the input and h(x) is the output. It is helpful to rewrite the formula as Output = h(Input) = (Input)2 − 3・ (Input) + 5. (a) For h(2), we have Input = 2, so h(2) = (2)2 − 3・(2) + 5 = 3. 2 Finding Output Values: Evaluating a Function Example 4 continued h(x) = x2 − 3x + 5. Evaluate (b) h(−3) (c) h(a) (d) h(a) − h(2) Solution (b) In this case, Input = −3. We substitute and simplify h(−3) = (−3)2 − 3(−3) + 5 = 9 + 9 + 5 = 23. (c) In this case, Input = a: h(a) = (a)2 − 3(a) + 5 = a2 − 3a + 5 . (d) Using h(2) = 3 from part (a): h(a) − h(2) = a2 − 3a + 5 − 3 = a2 − 3a + 2. 3 Finding Input Values: Solving Equations Example 6 (a) Suppose 1 f ( x) x4 (a) Find an x-value that results in f(x) = 2. Solution (a) To find an x-value that results in f(x) = 2, solve the equation 1 x4 1 4 x4 4( x 4) 1 2 Squaring both sides Multiplying both sides by x - 4 4 x 16 1 x 17 / 4 4.25 4 Finding Output Values from Tables Example 8(a) The table shows the revenue, R = f(t), received, by the National Football League, NFL, from network TV as a function of the year, t, since 1975. (a) Evaluate and interpret f(25). Year, t (since 1975) Revenue, R (million $) 0 5 10 15 20 25 30 201 364 651 1075 1159 2200 2200 Solution (a) The table shows f(25) = 2200. Since t = 25 in the year 2000, we know that NFL’s revenue from TV was $2200 million in the year 2000. 5 Finding Input/Output Values from Graphs Exercise 35 (a) and (c) An epidemic of influenza spreads through a city. The figure to the right shows the graph of I = f(w), where I is the number of individuals (in thousands) infected w weeks after the epidemic begins. 9 8 7 6 5 4 3 2 1 0 f f(w) 0 2 4 6 8 10 12 14 16 w (a) Evaluate f(2) and explain its meaning . (b) Solve f(w) = 4.5 and explain what the solutions means. Solution (a) f(2) ≈ 7, so 7 thousand individuals were infected 2 weeks after onset. (b) If you imagine a horizontal line drawn at f = 4.5, it would intersect the graph at two points, at approximately w = 1 and w = 10. This would mean that 4.5 thousand individuals were infected with influenza both 6 one week and ten weeks after the epidemic began.