The marketing manager of a company is planning to have a grand party for her boss. She found out that the cost to hire a caterer for the party
depends on the number of guests attending. If 100 people attend, the cost per person will be $500. For each person less than 100, the cost will increase by $75. Assume that no more than 100 people will attend. Let 𝒙 represents the number of guests who do not attend
(out of the 100 who are invited). For example, if 95 attend, 𝒙 = 5.
a. Write a function defined by 𝑁(𝑥) giving the number of guests.
n(x) = 100 - x
b. Write a function defined by 𝐺(𝑥) giving the cost per guest.
g(x) = 500 + 75 * x
c. Write and interpret the function defined by (𝑁 • 𝐺)(𝑥).
(n*g)(x) = n(x) * g(x) = (100 - x) * (500 + 75 * x)
this function gives the total cost based on the number of guests who attend.
if x = 0, then 100 people attend and the total cost is 500 * 100 = 50,000
if x = 40, then 60 people attend and the total cost is 60 * (500 + 75*40) which is equal to 60 * (3500) = 210,000.
if x = 100, the 0 people attended and the total cost is 0.
you can see this from the graph of the function (n*g)(x) shown below:
in the graph, (n*g)(x) is equal to y.
d. What is the total cost if 60 people attend?
the total cost is 60 people attended is 210,000.
that is shown on the graph.
it occurs when x = 40.
x is the number of people who do not attend.
if 40 people did not attend, that means that 60 people did attend.
Problem 2
What is the domain of the function 𝒇(𝒙) = −𝟒 / (√𝟐𝒙 + 𝟏)? Find the range using the concept of inverse function.
the function, as i understand it, is:
f(x) = -4 / (sqrt(x) + 1)
since you can't take the square root of a negative number, then x has to be greater than or equal to 0.
your domain is therefore all values of x greater than or equal to 0.
your range is therefore based on all values of x greater than or equal to 0.
when x = 0, the denominator is 1 and the value of the function is -4 divided by 1 which is equal to -4.
any value of x greater than 0 will result in a value of y greater than -4.
when x is 1 million, the value of y will be -4 / 1,000,001 which is a very small number very close to 0.
it appears that the value of y will approach 0 as x approaches positive infinity.
so your range will be y >= -4 and y < 0 which can be written as -4 <= y < 0.
the inverse function is x = -4 / sqrt(y) + 1
the domain of this function will be y greater than or equal to 0.
the range of this function will be x >= -4 and x <= 0.
if you try to solve the inverse function for y, you can run into some conceptual difficulties because the graph will show the domain as all real values of x except at x = 0.
you just have to remember that the domain of the original function becomes the range of the inverse function and the range of the original function becomes the domain of the inverse function.
since the domain of the original function is all x greater than or equal to 0, then the range of the inverse function is all y greater than or equal to 0.
since the range of the original function is all y greater than or equal to -4 and less than 0, then the domain of the inverse function is all x greater than or equal to -4 and less than 0.
the graph of the inverse function of x = -4 / (sqrt(y) + 1) shows this automatically.
the graph of the inverse function of y = (-4/x -1)^2 is correct for all values of x greater than or equal to -4 and less than 0, but not correct for all values of x greater than or equal to 0.
in that case, the graph is misleading and you have to use your concept of inverse functions to only look at the parts of the graph that make sense based on that.
here's the graph of the inverse function as x = -4 / (sqrt(y) + 1)
here's the graph of the inverse function after it was solved for y in terms of x.
note that the automatic restriction on the value of x being greater than or equal to 0 is not there anymore because it is not under the square root sign.
regardless, if you understand that the range of the inverse function is equal to the domain of the original function and that the domain of the inverse function is equal to the range of the original function, then you will always be able to see which part of the graph is legitimate and which part isn't.
if you were graphing the inverse function manually, you would not have shown the value of x greater than or equal to 0 because the range of the original function was y greater than or equal to 4 and y less than 0.
since the graphing software did not know that, it just went on its merry way to graph what it could.
it would be up to you to keep those restrictions in mind when graphing the inverse function once you had solved it for y.