Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
ChapterP: Prerequisites
SectionP.7: A Library Of Parent Functions
Problem 47E
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![Identify the correct steps involved in proving that x-1 < [x] ≤x≤ [x] <x+1, where x is a real number. (Check all that apply.)
Check All That Apply
We can write the real number x as [x]+e, where is a real number satisfying 0 sr<1.
Since E=x-[x], we have 0 sx-[x]<1. The first two inequalities x-1 < [x] and [x] ≤x follow algebraically.
Since t = x-[x], we have 0 sx-[x]<1. The first two inequalities x-1s [x] and [x] <x follow algebraically,
For the other two inequalities, we can write x= [x]-e, where again 0 se<1.
Using 0 [x]-x<1, we can write xs [xland [x] <x+1.It shows that x-1 < [x] sxs [x] <x+1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F52fad7f9-097d-4ed4-a9b8-ba0b37335f42%2F99a90238-7406-4a20-b2ae-27665b29cd44%2Fxn57n7k_processed.png&w=3840&q=75)
Transcribed Image Text:Identify the correct steps involved in proving that x-1 < [x] ≤x≤ [x] <x+1, where x is a real number. (Check all that apply.)
Check All That Apply
We can write the real number x as [x]+e, where is a real number satisfying 0 sr<1.
Since E=x-[x], we have 0 sx-[x]<1. The first two inequalities x-1 < [x] and [x] ≤x follow algebraically.
Since t = x-[x], we have 0 sx-[x]<1. The first two inequalities x-1s [x] and [x] <x follow algebraically,
For the other two inequalities, we can write x= [x]-e, where again 0 se<1.
Using 0 [x]-x<1, we can write xs [xland [x] <x+1.It shows that x-1 < [x] sxs [x] <x+1.
![Let /be a function from A to B and S be a subset of 8. Identify the correct steps involved in showing that f-¹ (S) = f-¹ (S). (Check all that apply)
Check All That Apply
We know that S=B-S and d f-¹ (S) = A-1-¹ (s).
We know that S=A-S and f-¹ (S) B-f
f-¹ (S)=B-f-¹ (s).
Let x Ef (S). This means that f(x) ES or equivalently that f(x) @S.
Therefore, by the definition of inverse image, xer¹(5): so, x Ef-¹ (S).
Therefore, by the definition of inverse image, xer¹(% 50, x Ef-¹ (S).
Assume that x Ef-¹ (S). Then, x(S). This means that ) S
It shows that f(x) ES. Therefore, by the definition of inverse image. x Ef-¹ (S).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F52fad7f9-097d-4ed4-a9b8-ba0b37335f42%2F99a90238-7406-4a20-b2ae-27665b29cd44%2Fxwx245q_processed.png&w=3840&q=75)
Transcribed Image Text:Let /be a function from A to B and S be a subset of 8. Identify the correct steps involved in showing that f-¹ (S) = f-¹ (S). (Check all that apply)
Check All That Apply
We know that S=B-S and d f-¹ (S) = A-1-¹ (s).
We know that S=A-S and f-¹ (S) B-f
f-¹ (S)=B-f-¹ (s).
Let x Ef (S). This means that f(x) ES or equivalently that f(x) @S.
Therefore, by the definition of inverse image, xer¹(5): so, x Ef-¹ (S).
Therefore, by the definition of inverse image, xer¹(% 50, x Ef-¹ (S).
Assume that x Ef-¹ (S). Then, x(S). This means that ) S
It shows that f(x) ES. Therefore, by the definition of inverse image. x Ef-¹ (S).
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