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UPSC NDA I 2016 Mathematics Solved Paper

Question Numbers: 70-71

Direction: For the next two (2) items that follow:

Let f(x) be the greatest integer function and g(x) be the modulus function.

Question : 70 of 120

Marks: +1, -0

What is

(g o f)(-\frac{5}{3})-(f o g)(-\frac{5}{3})

equal to?

Solution:

Concept:

Composite function:

(g \circ f)(x) = g(f(x))

Greatest integer function:

f(x)=[x]

gives the largest integer

\le x

Modulus function:

g(x)=|x|

gives the non‑negative value of

x

Explanation:

Given

f(x)=[x]

and

g(x)=|x|

First compute

(g \circ f)\left(-\frac{5}{3}\right)=g\left(f(-\frac{5}{3})\right)

f(-\frac{5}{3}) = [-1.666\ldots] = -2

Then

g(-2) = |-2| = 2

(g \circ f)(-\frac{5}{3}) = 2

Next compute

(f \circ g)(-\frac{5}{3})=f\left(g(-\frac{5}{3})\right)

g(-\frac{5}{3}) = |-\frac{5}{3}| = \frac{5}{3}

f(\frac{5}{3}) = [1.666\ldots] = 1

(f \circ g)(-\frac{5}{3}) = 1

Subtract:

2 - 1 = 1

Answer:

1

(Option C).

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