Let $y$ be the operation given by $y=4/y-y$ . Which of the following statements are true? I. If $0<y$ , then $y$ is negative. II. If $0<y<z$ , then $y>z$ . III. If $0<y$ then $y(y)$ is less than 5.

Answer Choices

Correct answer marked below

I only

II only

III only

II and III

Correct

I, II, and III

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Comments

NaderJun 19

This one took me a while. I tested y=1 and y=2 for statement I and both gave negative results, but then I realized I need to be careful with small fractions. For y=0.5, *y = 4/0.5 - 0.5 = 8 - 0.5 = 7.5, which is positive. So I is false. Then for II, I tried y=1, z=2: *1=3, *2=1, so 3>1, true. But is it always true? I think so because the function decreases as y increases for positive y. For III, I plugged y=1 and got 1*(3) = 3, less than 5; but y=0.5 gives 0.5*7.5 = 3.75, still less than 5. I thought maybe it's always true, but I'm not 100% sure. Ended up guessing D. Anyone else find this tricky?

AshwantJun 30

For III, try y=0.1: *0.1 = 39.9, then 0.1*39.9 = 3.99, still less than 5. And y=10 gives *10 = -9.6, 10*(-9.6) = -96, also less than 5. So it seems true.

SuryanshooJun 28

Yeah, I also got D. But I had to double-check II with calculus. The derivative of 4/y - y is -4/y^2 - 1, which is always negative for y>0, so it's strictly decreasing. So II is definitely true.

DukeJun 3

I got B. I is false because of y=0.5. II is true because the function is decreasing. For III, I thought it was false because if y is between 0 and 1, *y is large, and y times that might exceed 5? But then I computed y=0.5 gave 3.75, and y=0.9 gave 0.9*(4/0.9 - 0.9) = 0.9*(4.444... - 0.9) = 0.9*3.544... = 3.19. So it's less than 5. Maybe I made a mistake in my reasoning. Can someone explain why III is true?

RiyanshJun 17

You can simplify y(*y) = y(4/y - y) = 4 - y^2. Since y>0, y^2>0, so 4 - y^2 < 4, which is less than 5. So III is always true.

SeharJun 4

Ah, that's a neat simplification. I didn't see that. So III is true. Then D is correct.

MalothuJul 19

This is a solid 700-level question. The trick is to not assume the function is always negative for positive y. I initially thought I was true until I tried y=0.5. Then II is straightforward if you know it's decreasing. III is easy once you see the simplification. I got D after a few minutes.

MahnoorJun 28

I got E at first because I didn't test small fractions for I. Then I saw the comment about y=0.5 and felt silly. II and III are true, so D. Good question to remind me to always test edge cases.

What this question tests

What it tests

Your ability to evaluate and compose functions, including unusual defined operations.

Common trap

Applying function operations in the wrong order or ignoring domain restrictions.

Let ∗y*y∗y be the operation given by ∗y=4/y−y*y=4/y-y∗y=4/y−y. Which of the following statements are true? I. If 0<y0<y0<y, then ∗y*y∗y is negative. II. If 0<y<z0<y<z0<y<z, then ∗y>∗z*y>*z∗y>∗z. III. If 0<y0<y0<y then y(∗y)y(*y)y(∗y) is less than 5.