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.