Exercise 28 Page 272

We are given some information about a certain machine.

• Warm-up time: 10 min
• Running one cycle time: 15 min
• Operation time including warm-up: 6 hours

We are asked to draw a graph showing the total time the machine operates during a day as a function of the number of cycles it runs.

Domain and Range

Let t(x) be the total time in minutes, the machine operates to run x cycles of 15 minutes each. Recall that there is a warm-up time of 10 minutes. t(x)= 15x+ 10Note that x cannot be negative, since it is impossible to run a negative number of cycles. Furthermore, since the machine cannot run partial cycles, x is an integer. rcl & x & ↙ & & ↘ x≥ 0 & & Integer Now, we have that x≥ 0. Let's find the corresponding output of x=0 by substituting 0 for x in the above function rule.

We found that t(0)=10. This means that the machine will only run for 10 minutes if no cycles are run. Moreover, we know that the machine can operate for as long as 6hours a day, which is equivalent to 6* 60= 360 minutes. By substituting 360 for t(x), we can determine the maximum number of cycles that the machine can run.

Therefore, the maximum number of cycles the machine can run in 6 hours is 23, since partial cycles are not allowed. We can write the domain using the obtained values. Domain:& { x | x is an integer, 0 ≤ x ≤ 23} Let's construct a table to find the range values. Keep the domain in mind! Since the function is discrete, we must find the corresponding range for each value in our domain.

Using the table, we can write the range. Range: { &10,25,40,55,70,85,100, &115,130,145,160,175, &190,205,220,235,250,265, &280,295,310,325, 340,355 }

Is the Function a Linear Function?

Let's now plot the obtained points. Again, keep in mind domain and range!

Since the points appear to be on a straight line, the function is a linear function.