# CAT Practice : Functions

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The following topics are covered in the CAT quant section from Algebra: Functions. Detailed explanatory answers, solution videos and slide decks are also provided.
1. ### 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
2. ### 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
3. ### 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)?

• Correct AnswerDomain: ( -∞, +∞), Range -[0, ∞] Correct answer
• Compound functions
• Medium
4. ### 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
• Correct AnswerChoice (C).
Roughly 0.14
• Greatest Integer Function
• Hard
5. ### 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) = 2x
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
6. ### Domain and Range

How many elements are present in the domain of 9–xCx+1?

1. 5
2. 6
3. 4
4. 7
7. ### 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
• Correct AnswerChoice (B).
1/9
• Unconventional Question
• Medium
8. ### 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
9. ### 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
10. ### 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(fn - 1 (x)). What is f(3) + f2(-3) + f3(3) + f4(-3)?

1. -2/3
2. 14/3
3. 0
4. 3
• Correct AnswerChoice (B).
14/3
• Function of a function
• Medium
11. ### 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
12. ### Function of a function

The value of fogoh(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
13. ### 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)
14. ### 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)
15. ### 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
• Correct AnswerChoice (B).
• Greatest Integer Function
• Medium
16. ### 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
• Correct AnswerChoice (D).
• Greatest Integer Function
• Medium
17. ### 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
• Correct AnswerChoice (D).
• Greatest Integer Function
• Medium
18. ### 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))))))))
19. ### 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
20. ### 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
21. ### 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
22. ### Value of f(x)

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, f(2) = 4, find the value of f(4).

1. 8
2. 18
3. 25
4. 28
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