PSAT/NMSQT Prep 2022 - Eggert M.D., Strelka A. 2022

Describing real-life situations with functions
Functions
Passport to advanced math

Learning objective

After this lesson, you will be able to:

· Write a function to describe a rule or data set

To answer a question like this:

A certain meat distribution company sells several varieties of meat. The company sells the different varieties in differently sized packages. The number of pounds per package and the profit per pound for the different varieties is shown in the table above. The relationship between the number of pounds per package (m) and the profit, in dollars, that the company makes per pound (p) can be represented by a linear function. Which of the following functions correctly represents the relationship?

A. p(m) = 0.09m − 0.41

B. p(m) = 0.08m − 0.82

C. p(m) = 0.07m − 1.11

D. p(m) = 0.06m − 1.42

You need to know this:

Modeling real-life situations using functions is the same as modeling them using equations; the only difference is the function notation and the rule that each input has only one output.

For example, suppose a homeowner wants to determine the cost of installing a certain amount of carpet in her living room. Say that the carpet costs $0.86 per square foot, the installer charges a $29 installation fee, and sales tax on the total cost is 7%. Using your algebra and function knowledge, you can describe this situation in which the cost, c, is a function of square footage, f. The equation would be c = 1.07(0.86f + 29). In function notation, this becomes c(f) = 1.07(0.86f + 29), where c(f) is shorthand for “cost as a function of square footage.” The following table summarizes what each piece of the function represents in the scenario.

You need to do this:

In word problems involving function notation, translate the math equations exactly as you learned in chapter 4 in the Word Problems lesson, but substitute f(x) for y.

Explanation:

Note that the question asks for the relationship between the pounds of meat per package, m, and the profit per pound, p, and that the answer choices all start with p(m). Given the context, this must mean, “profit as a function of the pounds of meat.” All the choices express a linear relationship, so you can’t rule out any of them on that basis.

There are several approaches you could take to find the correct answer. One would be to recognize that all the choices are in the form p(m) = km + b (a variation of the slope-intercept form y = mx + b) and that you can set up a system of linear equations using the data from any two rows of the table to solve for k and b. That approach would look like this:

If k = 0.06 and b = −1.42, the correct function is p(m) = 0.06k − 1.42, so (D) is correct.

Another approach would be to use two of the pairs of data points from the table to calculate a slope; for example, using the “sausage” and “bacon” rows would yield Because only one answer has a slope of 0.06, you can pick (D).

One last approach: you could backsolve. Plug any one of the rows of data from the table into all four answer choices. The fourth row, “Bacon,” has the easiest numbers to work with, so use those. You are checking which equation will produce a profit of $0.50 per pound given 32 pounds per package:

Again, (D) is correct.

Try on Your Own

Directions: Take as much time as you need on these questions. Work carefully and methodically. There will be an opportunity for timed practice at the end of the chapter.

HINT: Are there any answer choices in Q11 that you can immediately eliminate?

11. A biologist studying the birth rate of a certain fish uses the function b(n) to analyze the fish’s effect on other parts of the ecosystem, where n is the number of eggs laid by the fish over a given period of time. Which of the following lists could represent the domain for the biologist’s function?

A. {... −1,500, −1,000, −500, 0, 500, 1,000, 1,500...}

B.{−1,500, −1,000, −500, 0, 500, 1,000, 1,500}

C.{0, 0.25, 0.5, 0.75, 1, 1.25, 1.5...}

D. {0, 500, 1,000, 1,500, 2,000...}

12. A book publisher pays writers a base fee of $2,500 for each book that it publishes, plus 5 cents per word. If one of its writers earned $8,000 on her book last year, how many words, w, did she write for the publisher?

A. 11,000

B.110,000

C.155,000

D. 250,000

HINT: How does knowing the starting height of the solution help you construct the function in Q13?

13. Tyree is dropping old pennies into a jar that contains a cleaning solution. As he adds more pennies, the height of the solution in the jar changes based on the number of pennies he adds. The figure shows this relationship after 50 pennies have been dropped in the jar. If the height of the solution in the jar was 5 inches (in) before any pennies were added, which of the following linear functions represents the relationship between the number of pennies, p, and the height in inches, h(p), of the solution in the jar?

A. h(p) = 0.7p + 5

B.h(p) = 0.7p + 8.5

C.h(p) = 0.07p + 5

D. h(p) = 0.07p + 8.5

HINT: Begin Q14 by calculating the parts per million at both 10 and 20 hours.

14. Doctors use the function shown above to calculate the concentration, in parts per million, of a certain drug in a patient’s bloodstream after t hours. How many more parts per million of the drug are in the bloodstream after 20 hours than after 10 hours?

15. A teacher is buying supplies for the upcoming school year. Every year, basic classroom supplies such as chalk and paper cost her $500, and she spends an additional $25 per child in her class. The school reimburses her $10 per child for half the children in her class. Which function best describes the amount, in dollars, that the teacher spends per school year on supplies, given that s represents the number of students in the class?

A.

B.

C.

D.