Reflect - Functions - Passport to advanced math

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

Reflect
Functions
Passport to advanced math

Directions: Take a few minutes to recall what you’ve learned and what you’ve been practicing in this chapter. Consider the following questions, jot down your best answer for each one, and then compare your reflections to the expert responses on the following page. Use your level of confidence to determine what to do next.

What are the domain and range of a function?





What is another way to write the function f(x) = x + 4?





In the function above, what does x represent? What does f(x) represent?





What will the above function look like when graphed?





In a function whose x-value represents time, what does the y-intercept represent?





Expert Responses

What are the domain and range of a function?

The domain of a function indicates the possible x -values, and the range of a function indicates the possible y -values. For example, in the function f(x)= x2 , the domain is all real numbers because any number can be squared, and the range is any number greater than or equal to 0, because x2 can’t be negative.

What is another way to write the function f(x) = x + 4?

When you graph the function on the xy -coordinate plane, you can replace f(x)with y . This function is equivalent to y= x+4.

In the function above, what does x represent? What does f(x) represent?

In this function, x is the input and f(x)is the output.

What will the above function look like when graphed?

The slope of the line is 1 and its y -intercept is 4, so it will move from the lower left to the upper right and cross the y -axis at y=4.

In a function whose x-value represents time, what does the y-intercept represent?

The y -intercept represents the initial quantity when t=0. Say a function represents the progress of a machine manufacturing widgets at a rate of 6 widgets per hour. The machine adds the widgets it makes to a growing pile that consisted of 12 widgets when the machine started working. If this function were graphed as a function of time, the y -intercept would be 12—the pile of 12 widgets that were there when the machine started its task.

Next Steps

If you answered most questions correctly in the “How Much Have You Learned?” section, and if your responses to the Reflect questions were similar to those of the PSAT expert, then consider Functions an area of strength and move on to the next chapter. Come back to this topic periodically to prevent yourself from getting rusty.

If you don’t yet feel confident, review those parts of this chapter that you have not yet mastered. All three lessons in this chapter cover question types that are fairly common on the PSAT, and it is to your advantage to have a firm grasp on this material, so go back over it until you feel more confident. Then, try the questions you missed again. As always, be sure to review the explanations closely.

Answers and Explanations

1. A

Difficulty: Easy

Getting to the Answer: You’re looking for a value that is NOT possible for f(x). You’re told f(x) = 0 for all x > −1, so you can eliminate (B) right away. The other part of the function gives f(x) = x2x for all x ≤ −1. Plugging any negative number less than or equal to −1 into this part of the function gives a negative number squared minus the original negative number, which will always be positive. For example, f (−3) = (−3)2 − (−3) = 9 + 3 = 12. Therefore, the range of f(x) will always be greater than or equal to 0. This eliminates (C) and (D), leaving (A) as the correct answer.

2. A

Difficulty: Easy

Getting to the Answer: Make sure you compute the functions in this composition in the correct order. Start with the innermost set of parentheses as you would with any composition of functions. Substitute 3 for x in f(x): image Next, take the output of f (3) and use it as the input for g(x): imageThis matches (A).

3. C

Difficulty: Easy

Getting to the Answer: Substitute 5 and then 2 for x in the function and find the difference in the results:

image

Choice (C) is correct. Caution—this is not the same as subtracting 5 − 2 and then substituting 3 into the function.

4. B

Difficulty: Medium

Strategic Advice: The notation (g(h(x)) can be read “g of h of x.” It means that when x is substituted into h(x), the output becomes the input for g(x).

Getting to the Answer: Start with the innermost function. Use the h(x) table to find h(−2); when x = −2, the output h(x) = −4. Use −4 as the input for g(x) to find g(−4) from the g(x) table. When x = −4, the output g(x) = 0. Choice (B) is correct.

5. A

Difficulty: Hard

Getting to the Answer: The notation f (g(x)) means: find g(x), then substitute the result into f(x). Substitute image for x in g(x) and simplify:

image

Next, substitute the result into f(x):

image

Therefore, image so (A) is correct.

6. D

Difficulty: Easy

Getting to the Answer: Begin by finding f (0). When x = 0 on the graph of function f, y = 4. Therefore, f (0) = 4. Now, repeat for image When image on function g, y = 10; thus, image You’re asked for the sum of f (0) and image so substitute the appropriate numbers to get 4 + 10 = 14, which is (D).

7. D

Difficulty: Easy

Getting to the Answer: To determine the domain, look at the x-values. To determine the range, look at the y-values. For the domain, the graph is continuous and has arrows on both sides, so the domain is all real numbers. This means you can eliminate choices (A) and (B). For the range, the function’s maximum is located at (0, 5), which means the highest possible value of f(x) is 5. The graph is continuous and opens downward, so the range of the function is f(x) ≤ 5, making (D) correct.

8. B

Difficulty: Medium

Getting to the Answer: Compare each answer choice, one at a time, to the graph. Be careful—you’re looking for the statement that is NOT true, so cross out true statements as you go. The statement in (A) is true (and therefore not correct) because at x = 0, y is also 0. For (B), when x = 2, q(x), or y, is somewhere between 1 and 2, not equal to 4, so this statement is NOT true, making (B) the correct answer. For the record, the statements in (C) and (D) are both true because the x-values (domain) and y-values (range) are all greater than or equal to 0.

9. A

Difficulty: Medium

Getting to the Answer: Graphically, the notation f (3) means the y-value when x = 3. It may help to draw dots on the graph. Find x = 3 on the x-axis, then mark each function at this x-value on the graph as shown here:

Now, read the y-coordinates from the graph, paying close attention to which function is which, and subtract: f (3) = −2 and g(3) = 1, so f (3) − g(3) = −2 − 1 = −3. (A) is correct.

10. 2

Difficulty: Hard

Getting to the Answer: First, find the maximum value, which means the highest y-value. The maximum value occurs at the point (0, 3) so 3 is the maximum value. Find f (3) by locating the y-value when x = 3. The point (3, −2) means that f(3) = −2. Remember that the question asks for the value of -f (z) = -f (3) = -(−2) = 2. Grid in 2.

11. D

Difficulty: Easy

Getting to the Answer: The domain of a function represents the possible input values. In this function, the input values are represented by n, which is the number of eggs laid by the fish over a given period of time. Because there cannot be a negative number of eggs laid or a fraction of an egg laid, the list in (D) is the only one that could represent the function’s domain.

12. B

Difficulty: Easy

Getting to the Answer: First, figure out the function that will show you how the magazine publisher pays its writers. The writers earn $2,500 plus $0.05 per word. So the function for earnings, E, is: E(w) = $2,500 + $0.05(w).

Plug in what you know and solve for what you don’t. The writer made $8,000 for writing one book, so $8,000 = $2,500 + .05(w). Solve for w:

$8,000 = $2,500 + $0.05(w)

$5,500 = $0.05(w)

w = 110,000

So to earn $8,000, the writer wrote 110,000 words for the publishing company. That matches (B).

13. C

Difficulty: Medium

Strategic Advice: The question tells you that the function is linear, which means you need to know the slope (rate of change in the height of the solution) and the y-intercept (height of the solution when there are 0 pennies) to pick the correct function.

Getting to the Answer: You already know the height of the solution when there are 0 pennies—it’s 5 inches. This means you can eliminate (B) and (D). To determine the rate of change in the height of the solution, write what you know as ordered pairs, and then use the slope formula:

At 0 pennies, the height is 5 inches → (0, 5).

At 50 pennies, the height is 8.5 inches → (50, 8.5).

The rate of change in the height of the solution is image This means the correct function is h(p) = 0.07p + 5, which is (C).

14. 5

Difficulty: Medium

Getting to the Answer: Evaluate the function at t = 20 and at t = 10, then subtract the results. Make sure you follow the correct order of operations as you simplify:

image

image

The question asks how many more parts per million are in a patient’s bloodstream after 20 hours than after 10 hours, so subtract 22 − 17 = 5. Grid in 5.

15. A

Difficulty: Medium

Getting to the Answer: The teacher spends $500 plus $25 per child, or 500 + 25s. But she is reimbursed $10 for half of the children in the class, so subtract 10(0.5)(s) to give 500 + 25s − 5s = 500 + 20s. The correct function is f(s) = 500 + 20s, which matches (A).

16. B

Difficulty: Medium

Category: Describing Real-Life Situations with Functions

Strategic Advice: When modeling a real-life situation with a linear function, the starting point in the description is the y-intercept of the equation and the rate of change is the slope. Eliminate choices as you go; you may find that you are able to answer the question after only one or two steps. Never do more math than necessary to answer the question.

Getting to the Answer: In this question, the agency is reducing the amount of toxic waste disposed annually, so the slope must be negative. Eliminate (C) and (D) because their positive slopes indicate an increasing function.

The starting point, or y-intercept, is the amount of toxic waste the company is now discarding, or 10.8 million pounds. Unfortunately, this value is the same in (A) and (B), so you’ll need to calculate the value of the slope more precisely.

The agency wants to eliminate all toxic waste produced in a year by the end of 14 years, so to find the amount of reduction per year, divide 10.8 by 14.

The value image does not appear in the choices, and the slopes in the choices do not have decimal points, so multiply the fraction by 1 in the form of image and reduce until you see a value that matches an answer choice. So, image so (B) is correct.

17. D

Difficulty: Medium

Category: Function Notation

Getting to the Answer: The question says that the value of f(x) = 5 and asks for the value of x that produces this result. This means you are solving for x, not substituting for x. Set the function equal to 5, and solve using inverse operations:

image

Thus, (D) is correct.

18. D

Difficulty: Hard

Category: Function Notation

Getting to the Answer: When dealing with a composition (or nested functions), the range of the inner function becomes the domain of the outer function, which in turn produces the range of the composition. In the composition f (g(x)), the function g(x) = x2 is the inner function. Every value of x, when substituted into this function, will result in a nonnegative value (because the value of x is squared). This means the smallest possible range value of g(x) is 0. Now, look at f(x): substituting large positive values of x in the function will result in large negative numbers. Consequently, substituting the smallest value from the range of g, which is 0, results in the largest range value for the composition, which is −0 + 5 = 5. Therefore, the range of f(g(x)) is ≤ 5. The only answer choice that does not fall within this range is (D).

19. B

Difficulty: Easy

Category: Graphs of Functions

Getting to the Answer: The domain of a function is all possible x-values. In this case, the arrows indicate the graph is going to continue infinitely in both directions, so every x-value is possible, meaning the domain of the function is all real numbers.(B) is correct.

Note that if you forgot whether it is the x− or y-value that gives you the domain, (A) would have been a tempting choice. (A) actually gives the correct range of the function, which is determined by all possible y-values.

20. 462

Difficulty: Medium

Category: Describing Real-Life Situations with Functions

Getting to the Answer: Questions sometimes include extra information that you don’t need to get the answer. For this question, all you need to consider is the difference in calories between playing basketball for 4 hours and playing basketball for 2 hours. Since that activity burns 231 calories per hour, the difference is 2 × 231 = 462. Grid in 462.

21. C

Difficulty: Easy

Category: Function Notation

Getting to the Answer: The trick to questions like this one is to make sure you’re careful with the order of the nested functions. Work from the inside out. In this case, you’re asked for g of f, so start with f(x), plugging in the given value for x: image Now, move to the outer function:

image This matches (C).

Note that if you had worked your nested functions in the wrong order—as you would if you’d been asked for f (g(x))—your final result would have been 6, which matches (A). Always double-check that you’re working your way through nested functions in the correct order, from the inside to the outside.

22. 81

Difficulty: Easy

Category: Function Notation

Getting to the Answer: Remember that f(x) means y. Substitute 3 for x and 6 for f(x) to find k:

image

Grid in 81.

23. D

Difficulty: Medium

Category: Describing Real-Life Situations with Functions

Getting to the Answer: The range of the function is all of the possible g(y) values. So this question asks you to determine which numbers are too small to be part of the range of the scientist’s data.

First, calculate how tall the trees would be when the scientist begins to use her function: h = 3.6 + 2.4(5) = 15.6 feet. Having solved for h, the function can now be written as g(y) = 15.6 + 2.4y, where y is greater than or equal to 0, and the range will be any value greater than or equal to 15.6. Only (D) is larger than 15.6, so it is correct.

24. A

Difficulty: Hard

Category: Function Notation

Getting to the Answer: To find f (g(x)), replace the variable in f(x) with g(x), or 2x − 1:

image

Thus, (A) is correct.

25. B

Difficulty: Medium

Category: Graphs of Functions

Getting to the Answer: Start by figuring out the y- values needed. On the graph,f (−2) has a value of 5, while g(2) has a value of 25. Therefore, f (−2) − g(2) = 5 − 25 = −20. This matches (B).

26. B

Difficulty: Medium

Category: Describing Real-Life Situations with Functions

Getting to the Answer: The cost for the first 100 feet will be $15(100) = $150. This number will be added to the cost for the remainder of the fence. Eliminate (C) and (D), which subtract the 150 instead of adding it.

Now, consider what happens after the first 100 feet. The remaining length of fencing is the total length, x, minus the first 100 feet, or x − 100. Multiply by 12 to find the cost for this section of the fence: 12(x − 100). The complete function, then, is c(x) = 150 + 12(x − 100). This matches (B).

27. C

Difficulty: Medium

Category: Graphs of Functions

Getting to the Answer: The domain is all the x-values that will yield a value for f(x). From left to right, notice that the graph goes from −3 to 3 and includes the endpoints. Thus, (C) is correct.