Ex: Find the Practical Domain of a Profit Function (Linear)

In this question, we’re asked to find the practical domain and practical range based
upon the given situation where it says you purchased
90 ice cream cones for $18 and plan to sell them at the park for 82 cents each, your
profit can be determined by the function p of x
equals zero point eight two x minus 18, where p represents your profit and x represents number
of ice cream cones sold. So the first thing we should recognize is that the profit function
is a linear function in slope intercept form,
meaning it fits the form f of x equals mx plus b, so notice how the y intercept or vertical intercept is negative 18 and the slope
is zero point eight two. So if you were to graph
the linear function on the coordinate plane,
it would look like this. If we ignored the
context of the situation, because the graph moves left and right without any holes or breaks, the domain would be all real numbers, and because the graph moves down and up indefinitely without any holes or break,
the range would also be all real numbers, but again,
because of the context of the situation, this would not be the practical domain and practical range. Let’s go back and look at our problem. Remember x or the input represents the number of ice cream cones sold and you purchased 90
ice cream cones for $18, so the least number of
cones you could sell would be zero, you can’t sell
a negative number of cones, and the most number of
cones you could sell would be 90 because that’s
how many you bought, which means the practical domain would be when x is greater
than or equal to zero and less that or equal to 90. For review, if they wanted to express the domain using interval notation, we’d use a square bracket
zero comma 90 square bracket. Remember, the square brackets indicate the end points are included. Now let’s go back to
our graph for a moment. Now that we know the practical domain, let’s go ahead and indicate
this on the x axis. The practical domain
is the closed interval from zero to 90, so of course, if we restrict the domain,
it’s going to affect the range. Notice how the function value of profit, when x is zero, would
be the y intercept here. This would be the lower
bound of the practical range. The upper bound of the
range would be the profit when x equals 90, which
would be approximately here. So this would be the practical range. Again, the lower bound is the y intercept, which we know is negative 18. To find the upper bound of the range, we’d have to substitute x equals 90 into our profit function. Let’s go ahead and show
how to find both of these. To find the lower bound
of the practical range, we have to find the
function value p of zero. Again, p of zero is going
to give us the y intercept. So this tells us that if you
sell zero ice cream cones, you’re going to lose
$18, because remember, that’s how much you paid to
purchase the 90 ice cream cones. And to find the upper bound,
we need to find the profit when you sell all 90 ice cream cones, so p of 90 would be equal
to zero point eight two times 90 minus 18, and I’ve
already determined this to save some time, it’s 55.8,
which does represent $55.80, but this would be the upper
bound of our practical range. So again, this does tell
us that if we sell zero ice cream cones, we’re actually going to have a loss of $18, if we sell all 90 ice cream cones, we’d have our maximum profit of $55.80. I hope you found this helpful.

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