SECTION 2.4 Library of Functions; Piecewise-defined Functions 107 Figure 52 y 8 (1, 21) (23, 7) (22, 5) (21, 3) 24 4 x 4 (2, 4) (1, 2) ( ) , 0 1 – 2 (0,1) (1, 1) The first potential x-intercept, x 1 2 , = satisfies the condition x 3 1, − ≤ < so x 1 2 = is an x-intercept. The second potential x-intercept, x 0, = does not satisfy the condition x 1, > so we discard it. The only x-intercept is 1 2 . The intercepts are 0, 1 ( ) and 1 2 , 0 . ( ) (d) To graph f, first graph the line y x2 1 = − + and keep only the part for which x 3 1. − ≤ < When x f x 1, 2, ( ) = = so plot the point 1, 2 . ( ) Finally, graph the parabola y x2 = and keep only the part for which x 1. > See Figure 52. Notice we use open circles at the points 1, 1 ( ) − and 1, 1 ( ) to indicate these points are not part of the graph. (e) From the graph, we conclude that the range of f is y y 1 , { } >− or the interval 1, . ( ) − ∞ Now Work PROBLEM 31 Cost of Electricity In the spring of 2023, Florida Power & Light (FPL) supplied electricity to residences in Florida for a monthly customer charge of $9.48 plus ( ) 7.063 cents per kilowatt-hour kWh for the first 1000 kWh supplied in the month and 8.055 cents per kWh for all usage over 1000 kWh in the month. Source: FPL, 2023. (a) What is the charge for using 500 kWh in a month? (b) What is the charge for using 1500 kWh in a month? (c) If C is the monthly charge for x kWh, develop a model relating the monthly charge and the number of kilowatt-hours used; that is, express C as a function of x. Solution EXAMPLE 5 (a) For 500 kWh, the charge is $9.48 plus 7.063 cents $0.07063 per kWh. ( ) = That is, Charge $9.48 $0.07063 500 $44.80 = + ⋅ = (b) For 1500 kWh, the charge is $9.48 plus 7.063 cents per kWh for the first 1000 kWh plus 8.055 cents per kWh for the 500 kWh in excess of 1000.That is, Charge $9.48$0.070631000$0.08055500$120.39 = + ⋅ + ⋅ = (c) Let x represent the number of kilowatt-hours used. If x 0 1000, ≤ ≤ then the monthly charge C (in dollars) can be found by multiplying x times $0.07063 and adding the monthly customer charge of $9.48. So, if x 0 1000, ≤ ≤ then C x x x 0.07063 9.48 0 1000 ( ) = + ≤ ≤ For x 1000, > the charge is x 0.07063 1000 9.48 0.08055 1000 ( ) ⋅ + + − because x 1000 ( ) − equals the usage in excess of 1000 kWh, which costs $0.08055 per kWh. That is, if x 1000, > then C x x x x x 0.07063 1000 9.48 0.08055 1000 70.63 9.48 0.08055 80.55 0.08055 0.44 1000 ( ) ( ) = ⋅ + + − = + + − = − > The rule for computing C follows two equations: ( ) = + ≤ ≤ − > ⎧ ⎨ ⎪⎪ ⎩⎪⎪ C x x x x x 0.07063 9.48 if 0 1000 0.08055 0.44 if 1000 The Model See Figure 53 for the graph. Note that both graphs are lines, but they have different slopes (rates) and intersect at the point 1000, 80.11 . ( ) Figure 53 y 180 160 140 120 100 80 60 40 20 500 1000 1500 x (0, 9.48) (500, 44.78) (1500, 120.39) (1000, 80.11) Usage (kWh) Charge (dollars)
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