Answer:
(a) C(t) = 0.99 +(0.07/2)(t -20 + |t -20|)
(b) domain: t > 0; range: C ≥ 0.99 (dollars)
(c) $0.99
(d) $3.79
Explanation:
You want an equation, its domain and range, and some values for a function that represents a calling plan that charges 99 cents for the first 20 minutes and 7 cents for each additional minute.
(a) Function
The scenario defines a piecewise function. It can be defined that way as ...
Here, C(t) is in dollars, and t is in minutes. You will note that this also covers the 0-minute case. We don't think that's really necessary, as a call will not be registered as a call if it has no duration.
Alternatively, the absolute value function can be used to write this on one line as ...
(b) Domain and Range
For the piecewise function defined above, the domain is x ≥ 0. The corresponding range is ...
(C = 0) ∪ (C ≥ 0.99) . . . . . includes the point (t, C) = (0, 0)
If we only consider calls of non-zero duration, then ...
- domain: t > 0
- range: C ≥ 0.99
(c) 12 minutes
A call lasting 12 minutes gets the rate for calls under 20 minutes.
cost for 12 minutes = $0.99
(d) 1 hour
A call lasting 60 minutes gets charged $0.99 for the first 20 minutes, and 7 cents for each of the final 40 minutes Its cost is the sum of these charges:
cost for 1 hour = $0.99 + 2.80 = $3.79
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Additional comment
The use of the absolute value function was suggested by a math app that didn't support the "max" function for variable expressions. We might have written the equation as ...
C(t) = 0.99 +0.07·max(0, t-20) . . . . t > 0
Whether you include a definition of C(t) for t=0 is your choice (and your grader's). We consider it for completeness, but we note that complicates the description of range. The absolute value (or max) function would give a $0.99 charge for a 0-minute call, which is probably incorrect. It is easier to avoid considering 0-minute calls.
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