🎉Ace the Final Stretch with our Last Mile Excellence – Your Ultimate CAT 2024 Boost!

CAT Questions | CAT Algebra Questions

CAT Quantitative Aptitude | CAT Algebra: Functions Questions

A CAT Algebra question from Functions that appears in the Quantitative Aptitude section of the CAT Exam consists of concepts from Algebra. Types of functions, Compound function, Greatest integer functions and domain and range of a function all appear in Functions. In CAT Exam, one can generally expect to get 1~2 questions from Functions. Make use of 2IIMs Free CAT Questions, provided with detailed solutions and Video explanations to obtain a wonderful CAT score. If you would like to take these questions as a Quiz, head on here to take these questions in a test format, absolutely free.

CAT Functions - Onto Functions

How many onto functions can be defined from the set A = {1, 2, 3, 4} to {a, b, c}?
1. 81
2. 79
3. 36
4. 45
Choice C
36
CAT Functions - Quadratic Expressions

Find the maximum value of f(x); if f(x) is defined as the Min {-(x – 1)² + 2, (x – 2)² + 1}
1. 1
2. 2
3. 0
4. 3
Choice B
2
CAT Functions - Compound functions

Consider functions f(x) = x² + 2x, g(x) = $\sqrt {{\rm{x + 1}}}$ and h(x) = g(f(x)). What are the domain and range of h(x)?

Domain: ( -∞, +∞), Range -[0, ∞)
CAT Functions - Greatest Integer Function

[x] = greatest integer less than or equal to x. If x lies between 3 and 5, 5 inclusive, what is the probability that [x²] = [x]²?
1. Roughly 0.64
2. Roughly 0.5
3. Roughly 0.14
4. Roughly 0.36
Choice C
Roughly 0.14
CAT Functions - Domain and Range

Give the domain and range of the following functions:
1. f(x) = x² + 1
2. g(x) = log(x + 1)
3. h(x) = 2^x
4. f(x) = 1/(x+1)
5. p(x) = |x + 1|
6. q(x) = [2x], where [x] gives the greatest integer less than or equal to x
CAT Functions - Domain and Range

How many elements are present in the domain of ^9–xC_x+1?
1. 5
2. 6
3. 4
4. 7
Choice B
6
CAT Functions - Unconventional Question

f(x + y) = f(x)f(y) for all x, y, f(4) = + 3 what is f(–8)?
1. 1/3
2. 1/9
3. 9
4. 6
Choice B
1/9
CAT Functions - Value of p - q

If f(x – 3) = 2x³ + p – qx and f(x² – 4) = x² – 8q + 6p, then what is the value of p – q?
1. 5
2. 10
3. 6
4. Cannot determine
Choice B
10
CAT Functions - Value of a

Given that x is real and f(x) = f(x + 1) + f(x – 1). Determine the value of ‘a’ that will satisfy f(x) + f(x + a) = 0?
1. -1
2. -2
3. 1
4. 3
Choice D
3
CAT Functions - Function of a function

x is a real number such that f(x) = 1/x when x > 0 and f(x) = 1/(x + 1) otherwise. Also fⁿ(x) = f(f^{n - 1}(x)). What is f(3) + f²(-3) + f³(3) + f⁴(-3)?
1. -2/3
2. 14/3
3. 0
4. 3
Choice B
14/3
CAT Functions - Identical Functions

Which of the following functions are identical? f(x) = \\frac{x^3}{x^2}\\)
g(x) = (√x)²
h(x) = x
1. f(x) and g(x)
2. f(x) and h(x)
3. All 3 are identical
4. None of these are identical
Choice D
None of these are identical
CAT Functions - Function of a function

The value of f∘g∘h(9) could be, if
f(x) = \\frac{1}{x}\\)
g(x) = \\frac{1}{(x-2)}\\)
h(x) = √x
1. 3
2. \\frac{1}{3}\\)
3. -5
4. None of these
Choice D
None of these
CAT Functions - Function of a function

For this question, assume the following operators: A*B = A² - B²
A-B = \\frac{A}{B}\\)
A+B = A * B
\\frac{A}{B}\\) = A+B
Which of the following expression would yield the result as x subtracted by y?
1. (x*y)-(x+5)
2. (\\frac{x}{y}\\))*(x-y)
3. (x*y) - (\\frac{x}{y}\\))
4. (x+y)*(x-y)
Choice C
(x*y) - (\\frac{x}{y}\\))
CAT Functions - Domain of Function

Find the domain of: \\frac{1}{(1-log (9-x))}\\) + √(x+1)?
1. (-∞,9)
2. [-1,9)
3. [-1,9) excluding 0
4. (-1,9)
Choice D
(-1,9)
CAT Functions - Value of a Sequence

If [X] – Greatest integer less than or equal to x. Find the value of
[√1] + [√2] + [√3] +……………………………………………………+ [√100]
1. 615
2. 625
3. 5050
4. 505
Choice B
625
CAT Functions - Greatest Integer Function

Find the value of x for which x[x] = 39
1. 6.244
2. 6.2
3. 6.3
4. 6.5
Choice D
6.5
CAT Functions - Greatest Integer Function

Find the value of x for which x[x] = 15
1. 3.5
2. 5
3. 6.1
4. None of these
Choice D
None of these
CAT Functions - Composite Function

If f(x) = \\frac{1}{g(x)}\\), then which of the following is correct?
1. f(f(g(g(f(x))))) = g(f(g(g(g(x)))))
2. f(f(f(g(g(g(f(g(x)))))))) = g(g(g(g(f(g(f(f(x))))))))
3. f(f(g(f(x)))) = g(g(f(g(x))))
4. f(g(f(f(g(f(g(g(x)))))))) = g(g(g(g(f(f(f(f(x))))))))
Choice D
f(g(f(f(g(f(g(g(x)))))))) = g(g(g(g(f(f(f(f(x))))))))
CAT Functions - Invertible Functions

If f(x) = \\frac{(x + 6)}{(x+2)}\\). Find the value of x for which f(x) = f^-1(x)?
1. -3
2. 2
3. Both A and B
4. None of these
Choice C
Both A and B
CAT Functions - Minimum Value of f(x)

If f(x) = |x| + |x+3| + |x+6| + ……………………………..+ |x+3t|, where x is an integer and t is a positive integer, find the minimum value of f(x) when t = 6
1. 63
2. 36
3. 30
4. 25
Choice B
36
CAT Functions - Minimum Value of f(x)

In the previous question if t = 7, for how many values of x, f(x) will be minimum?
1. 1
2. 2
3. 4
4. 8
Choice C
4
CAT Functions - Value of a Sequence

If \\frac{f(x)}{f(x-1)}\\) = \\frac{(x-2)}{(x+1)}\\), for all x ≥ 0 and f(x) is a positive-valued function and f(6) = 81, find the value of f(4)
1. 8
2. 283.5
3. 25
4. 28
Choice B
283.5

The Questions that follow, are from actual CAT papers. If you wish to take them separately or plan to solve actual CAT papers at a later point in time, It would be a good idea to stop here.

CAT 2023 Slot 3 - QA

Suppose $f(x, y)$ is a real-valued function such that $f(3 x+2 y, 2 x-5 y)=19 x$, for all real numbers $x$ and $y$. The value of $x$ for which $f(x, 2 x)=27$, is
3
CAT 2022 Slot 3 - QA

The minimum possible value of $\frac{x^2-6 x+10}{3-x}$, for $x\lt3$, is
1. $\frac{1}{2}$
2. $-\frac{1}{2}$
3. $2$
4. $-2$
Choice C
$2$
CAT 2022 Slot 3 - QA

Let $r$ be a real number and $f(x)=\left\{\begin{array}{cl}2 x-r & \text { if } x \geq r \\ r & \text { if } x\ lt r\end{array}\right.$. Then, the equation $f(x)=f(f(x))$ holds for all real values of $x$ where
1. $x \leq r$
2. $x \geq r$
3. $x \gt r$
4. $x \neq r$
Choice A
$x \leq r$
CAT 2022 Slot 2 - QA

Suppose for all integers $x$, there are two functions $f$ and $g$ such that $f(x)+f(x-1)-1=0$ and $g(x)=x^2$. If $f\left(x^2-x\right)=5$, then the value of the sum $f(g(5))+g(f(5))$ is
12
CAT 2022 Slot 2 - QA

Let $f(x)$ be a quadratic polynomial in $x$ such that $f(x) \geq 0$ for all real numbers $x$. If $f(2)=0$ and $f(4)=6$, then $f(-2)$ is equal to
1. 12
2. 36
3. 24
4. 6
Choice C
24
CAT 2022 Slot 1 - QA

The largest real value of $a$ for which the equation $|x+a|+|x-1|=2$ has an infinite number of solutions for $x$ is
1. -1
2. 0
3. 1
4. 2
Choice C
1
CAT 2022 Slot 1 - QA

For any real number $x$, let $[x]$ be the largest integer less than or equal to $x$. If $\sum_{n=1}^N\left[\frac{1}{5}+\frac{n}{25}\right]=25$ then $N$ is
44
CAT 2022 Slot 1 - QA

Let $0 \leq a \leq x \leq 100$ and $f(x)=|x-a|+|x-100|+|x-a-50|$. Then the maximum value of $f(x)$ becomes 100 when $a$ is equal to
1. 100
2. 25
3. 0
4. 50
Choice D
50
CAT 2021 Slot 3 - QA

If $f(x)=x^{2}-7 x$ and $g(x)=x+3$, then the minimum value of $f(g(x))-3 x$ is
1. - 20
2. - 15
3. - 12
4. - 16
Choice D
- 16

Correct: 43.86%
Incorrect: 19.68%
Unattempted: 36.46%
CAT 2021 Slot 2 - QA

For all real values of x, the range of the function f(x) = $\frac{x^{2}+2 x+4}{2 x^{2}+4 x+9}$ is
1. [ $\frac{3}{7}$ , $\frac{8}{9}$ )
2. [ $\frac{4}{9}$ , $\frac{8}{9}$ ]
3. [ $\frac{3}{7}$ , $\frac{1}{2}$ )
4. ( $\frac{3}{7}$ , $\frac{1}{2}$ )
Choice C
[ $\frac{3}{7}$ , $\frac{1}{2}$ )

Correct: 14.16%
Incorrect: 12.91%
Unattempted: 72.93%
CAT 2021 Slot 1 - QA

f(x) = $\frac{x^{2}+2 x-15}{x^{2}-7 x-18}$ is negative if and only if
1. -5 < x < -2 or 3 < x < 9
2. -2 < x < 3 or x > 9
3. x < -5 or 3 < x < 9
4. x < -5 or -2 < x < 3
Choice A
-5 < x < -2 or 3 < x < 9

Correct: 48.72%
Incorrect: 15.45%
Unattempted: 35.83%
CAT 2020 Question Paper Slot 3 - Functions

If f(x+y) = f(x)f(y) and f(5) = 4, then f(10) - f(-10) is equal to
1. 3
2. 0
3. 14.0625
4. 15.9375
CAT 2020 Question Paper Slot 2 - Functions

Let f(x) = x² + ax + b and g(x) = f(x + 1) - f(x - 1). If f(x) ≥ 0 for all real x, and g(20) = 72, then the smallest possible value of b is
1. 16
2. 1
3. 4
4. 0
CAT 2020 Question Paper Slot 1 - Functions

If f(5 + x) = f(5 - x) for every real x and f(x) = 0 has four distinct real roots, then the sum of the roots is
1. 0
2. 40
3. 10
4. 20
CAT 2019 Question Paper Slot 2 - Functions

Let f be a function such that f (mn) = f (m) f (n) for every positive integers m and n. If f (1), f (2) and f (3) are positive integers, f (1) < f (2), and f (24) = 54, then f (18) equals [TITA]
12
CAT 2019 Question Paper Slot 1 - Functions

For any positive integer n, let f(n) = n(n + 1) if n is even, and f(n) = n + 3 if n is odd. If m is a positive integer such that 8f(m + 1) - f(m) = 2, then m equals [TITA]
10
CAT 2019 Question Paper Slot 1 - Functions

Consider a function f(x+y) = f(x) f(y) where x , y are positive integers, and f(1) = 2. If f (a+1) + f (a+2) + ..... + f(a+n) = 16 (2ⁿ - 1) then a is equal to. [TITA]
3
CAT 2018 Question Paper Slot 2 - Functions

Let f(x)=max{5x, 52 - 2x²}, where x is any positive real number.Then the minimum possible value of f(x) is (TITA)
20
CAT 2018 Question Paper Slot 1 - Functions

If f(x + 2) = f(x) + f(x + 1) for all positive integers x, and f(11) = 91, f(15) = 617, then f(10) equals. [TITA]
54
CAT 2018 Question Paper Slot 1 - Functions

Let f(x)=min{2x², 52 - 5x}, where x is any positive real number.Then the maximum possible value of f(x) is [TITA]
32
CAT 2017 Question Paper Slot 2 - Functions

Let f(x) = x² and g(x) = 2^x, for all real x. Then the value of f(f(g(x)) + g(f(x))) at x = 1 is
1. 16
2. 18
3. 36
4. 40
Choice C
36
CAT 2017 Question Paper Slot 2 - Functions

If f(ab) = f(a)f(b) for all positive integers a and b, then the largest possible value of f(1) is [TITA]
1
CAT 2017 Question Paper Slot 2 - Functions

Let f(x) = 2x – 5 and g(x) = 7 – 2x. Then |f(x) + g(x)| = |f(x)| + |g(x)| if and only if
1. \\frac{5}{2}) < x < \\frac{7}{2})
2. x ≤ \\frac{5}{2}) or x ≥ \\frac{7}{2})
3. x < \\frac{5}{2}) or x ≥ \\frac{7}{2})
4. \\frac{5}{2}) ≤ x ≤ \\frac{7}{2})
Choice D
\\frac{5}{2}) ≤ x ≤ \\frac{7}{2})
CAT 2017 Question Paper Slot 1 - Functions

If f₁(x) = x² + 11x + n and f₂(x) = x, then the largest positive integer n for which the equation f₁(x) = f₂(x) has two distinct real roots, is: (TITA)
24
CAT 2017 Question Paper Slot 1 - Functions

If f(x) = \\frac{5x + 2}{3x - 5}) and g(x) = x² – 2x – 1, then the value of g(f(f(3))) is:
1. 2
2. \\frac{1}{3})
3. 6
4. \\frac{2}{3})
Choice A
2

The Questions that follow, are from actual XAT papers. If you wish to take them separately or plan to solve actual XAT papers at a later point in time, It would be a good idea to stop here.

XAT 2020 Question Paper - QADI
If A ʘ B = (A + B) × B, then what is (5 ʘ 2) ʘ 5 ?
1. 95
2. 275
3. 125
4. 74
5. 200
Choice A
95
XAT 2019 Question Paper - QADI
Consider the function f(x) = (x + 4)(x + 6)(x + 8) ⋯ (x + 98). The number of integers x for which f(x) < 0 is:
1. 23
2. 26
3. 24
4. 48
5. 49
Choice C
24

The Questions that follow, are from actual IPMAT papers. If you wish to take them separately or plan to solve actual IPMAT papers at a later point in time, It would be a good idea to stop here.

IPMAT 2020 Sample Paper - IPM Rohtak Quants
If minimum value of f(x) = x² + 2bx + 2c² is greater than the maximum value of g(x) = -x² - 2cx + b², then for real value of x.
1. |c| > √2|b|
2. √2|c| > b
3. 0 < c < √2b
4. no real value of a
Choice A
|c| > √2|b|
IPMAT 2020 Sample Paper - IPM Rohtak Quants
The set of all real numbers x for which x² - |x + 2 |+ x > 0, is
1. (-∞, -2) ∪ (2, ∞)
2. (-∞, -√2) ∪ (√2, ∞)
3. (-∞, -1) ∪ (1, ∞)
4. (√2, ∞)
Choice B
(-∞, -√2) ∪ (√2, ∞)
IPMAT 2020 Question Paper - IPM Indore Quants
The minimum value of f(x)=|3-x|+|2+x|+|5-x| is equal to __________.
7
IPMAT 2020 Question Paper - IPM Indore Quants
If \\frac{1}{1^{2}}) + \\frac{1}{2^{2}}) + \\frac{1}{3^{2}}) + .... upto ∞ = \\frac{π^{2}}{6}), then the value of \\frac{1}{1^{2}}) + \\frac{1}{3^{2}}) + \\frac{1}{5^{2}}) + .... upto ∞ is
1. \\frac{π^{2}}{8})
2. \\frac{π^{2}}{16})
3. \\frac{π^{2}}{12})
4. \\frac{π^{2}}{36})
Choice A
\\frac{π^{2}}{8})
IPMAT 2020 Question Paper - IPM Indore Quants
Given f(x) = x² + log₃x and g(y) = 2y + f(y), then the value of g(3) equals
1. 16
2. 15
3. 25
4. 26
Choice A
16
IPMAT 2019 Question Paper - IPM Indore Quants
A real-valued function f satisfies the relation f(x)f(y) = f(2xy + 3) + 3f(x + y) - 3f(y) + 6y, for all real numbers x and y, then the value of f(8) is
19
IPMAT 2019 Question Paper - IPM Indore Quants
For all real values of x, \\frac{3x^{2} - 6x + 12}{x^{2} + 2x + 4}\\) lies between 1 and k, and does not take any value above k. Then k equals
9
IPMAT 2019 Question Paper - IPM Indore Quants
The maximum value of the natural number n for which 21ⁿ divides 50! is
1. 6
2. 7
3. 8
4. 9
Choice C
8
IPMAT 2019 Question Paper - IPM Indore Quants
The function f(x) = \\frac{x^{3} - 5x^{2} - 8x}{3}\\) is
1. positive and monotonically increasing for x \\in (-\infty, \frac{5-\sqrt{57}}{2}\\)) and x \\in (\frac{5+\sqrt{57}}{2}, +\infty\\))
2. negative and monotonically decreasing for x \\in (-\infty, \frac{5-\sqrt{57}}{2}\\) and x \\in (\frac{5+\sqrt{57}}{2},+\infty\\))
3. negative and monotonically increasing for x \\in (-\infty, \frac{5-\sqrt{57}}{2}\\)) and positive and monotonically increasing for x \\in (\frac{5+\sqrt{57}}{2},+\infty\\))
4. positive and monotonically increasing for x \\in (-\infty, \frac{5-\sqrt{57}}{2}\\)) and negative and monotonically decreasing for x \\in (\frac{5+\sqrt{57}}{2},+\infty\\))
Choice C
negative and monotonically increasing for x \\in (-\infty, \frac{5-\sqrt{57}}{2}\\)) and positive and monotonically increasing for x \\in (\frac{5+\sqrt{57}}{2},+\infty\\))
IPMAT 2019 Question Paper - IPM Indore Quants
For a > b > c > 0, the minimum value of the function f(x) = |x - a| + |x - b| + |x - c| is
1. 2a - b - c
2. a + b - 2c
3. a + b + c
4. a - c
Choice D
a - c