Functions

Question 4

F(x) = x - [x], where [x] is the largest integer lesser than or equal to x, g(x) = x + abs(x), then

1. The union of the range of f(x) and g(x) is (0,1)
2. Neither f(x) nor g(x) can be negative for any x
3. g(x) + g(-x) = 2x

1. All three are true
2. b and c are true
3. Only b is true
4. Only c is true

Correct Answer is (b) and (c) are true. Choice (2)

Explanatory Answer

x > [x], so f(x) is always positive.

So, x - [x] > 0.

Since [x] is the largest integer lower than x, x - [x] is always less than 1.
So, f(x) lies between 0 and 1.

g(x) = 2x when x is positive and 0 when x is negative.

So, g(x) can only be positive, but it can take any positive value.
So, statement (a) cannot be true.

f(x) and g(x) both have to be greater than or equal to zero, so neither can be negative.
So, statement (b) is also true.

g(x) + g(-x) = 2x.
Statement (c) is als true.

Statement (b) and (c) are true. Answer choice (2).

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AIEEE Math Problems, Practice Questions and Answers : Listed Topicwise

Sets, Relations & Functions		Permutation Combination		Sequences & Series
Limits, Continuity & Differentiability		Statistics & Probability		Trigonometry
Mathematical Reasoning