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CAT 2025 Lesson : Functions - Box & Composite Functions

These are also called step functions as they look like steps when graphically plotted.

Greatest integer function of $x$ , also denoted as $\bm{[x]}$ , returns the greatest integer less than or equal to $x$ . For instance, $[1] \space = \space 1 \space , \space [1.2] \space = 1 \space , \space [1.99] \space = \space 1, [2] \space = \space 2$ This is also called a floor function.
Least integer function of $x$ , returns the least integer greater than or equal to $x$ . For instance, LIF( $1$ ) $= \space 1 \space , \space$ LIF( $1.2$ ) $\space =\space 2 \space , \space$ LIF( $1.99$ ) $\space = \space 2$ , LIF( $2$ ) $\space = \space 2$ This is also called a ceiling function.

If [X] denotes the greatest integer less than or equal to X, then

\left[ \dfrac{1}{3} \right] + \left[ \dfrac{1}{3} + \dfrac{1}{99} \right] + \left[ \dfrac{1}{3} + \dfrac{2}{99} \right] + \left[ \dfrac{1}{3} + \dfrac{98}{99} \right] =

[XAT 2008]

(

1

)33 (

2

)34 (

3

)66 (

4

)67 (

5

)98

\left[ \dfrac{1}{3} + \dfrac{0}{99} \right] = 0

\left[ \dfrac{1}{3} + \dfrac{1}{99}\right] = 0

and so on till

\left[ \dfrac{1}{3} + \dfrac{65}{99} \right]

From

\left[ \dfrac{1}{3} + \dfrac{66}{99} \right] = 1

onwards, all the terms equal

1

.

Between

66

and

98

(both inclusive), there are total of

98 - 66 + 1 = 33

terms, each of whose value is

1

\therefore

Total of these

33

terms is

33

.

Answer: (1)

33

In nested functions, we will given a function

f(x)

and asked to find

f(f(x))

. This is a function within a function.

In composite functions, we will be given

2

or more functions, say

f(x)

and

g(x)

. We will then be asked to find

f(g(x))

, etc.

f(x) = x^2, g(x) = 10 - x

and

h(x) = \dfrac{10}{x}

, then

(I) What is

f(g(x))?

(II) What is

g(g(x))?

(III) What is

h(g(f(x)))

Case I:

f(g(x)) = f(10 - x)

= (10 - x)^2

= \bm{100 - 20x + x^2}

Case II:

g(g(x)) = g(10 - x)

= 10 - (10 - x)

=x

Case III:

h(g(f(x))) = h(g(x^2))

= h(10 - x^2)

\bm{= \dfrac{10}{10 - x^2}}

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