Question 1: How many onto functions can be defined from the set A = {1, 2, 3, 4} to {a, b, c}?
A. 81
B. 79
C. 36
D. 45
Answer. 36
Explanation.
First let us think of the number of potential functions possible. Each element in A has three options in the codomain. So, the number of possible functions = 3^{4} = 81.
Now, within these, let us think about functions that are not onto. These can be under two scenarios.
Scenario 1: Elements in A being mapped on to exactly two of the elements in B (There will be one element in the codomain without a preimage).
Let us assume that elements are mapped into A and B. Number of ways in which this can be done = 2^{4} – 2 = 14
2^{4} because the number of options for each element is 2. Each can be mapped on to either A or B
2 because these 2^{4} selections would include the possibility that all elements are mapped on to A or all elements being mapped on to B. These two need to be deducted.
The elements could be mapped on B & C only or C & A only. So, total number of possible outcomes = 14 * 3 = 42.
Scenario 2: Elements in A being mapped to exactly one of the elements in B. (Two elements in B without preimage). There are three possible functions under this scenario. All elements mapped to a, or all elements mapped to b or all elements mapped to c.
Total number of onto functions = Total number of functions – Number of functions where one element from the codoamin remains without a preimage  Number of functions where 2 elements from the codoamin remain without a preimage.
⇒ Total number of onto functions = 81 – 42 – 3 = 81 – 45 = 36
The question is "How many onto functions can be defined from the set A = {1, 2, 3, 4} to {a, b, c}?"
Choice C is the correct answer.
Question 2: Find the maximum value of f(x); if f(x) is defined as the Min {(x – 1)^{2} + 2, (x – 2)^{2} + 1}
A. 1
B. 2
C. 0
D. 3
Answer. 2
Explanation.
First let us find the range where Min ((x – 1)^{2} + 2, (x – 2)^{2} + 1) is – (x – 1)^{2} + 2.
In other words, in which range is – (x – 1)^{2} + 2 < (x – 2)^{2} + 1.
–(x^{2} – 2x +1) + 2 < x^{2} – 4x + 4 + 1
0 < 2x^{2} – 6x + 4
x^{2} – 3x + 2 > 0
(x – 1) (x – 2) > 0
⇒ x > 2 or x < 1
So, for x ∈ (1, 2) , f(x) = (x – 2)^{2} + 1
And f(x) = –(x – 1)^{2} + 2 elsewhere.
Let us also compute f(1) and f(2)
f(1) = 2, f(2) = 1
For x ∈ (∞, 1), f(x) = –(x – 1)^{2} + 2
f(1) = 2
For x ∈ (1, 2), f(x) = (x – 2)^{2} + 1
f(2) = 1
For x ∈ (2, ∞), f(x) = –(x – 1)^{2} + 2
For x < 1 and x > 2, f(x) is (square) + 2 and so is less than 2.
When x lies between 1 and 2, the maximum value it can take is 2. f(1) = 2 is the highest value f(x) can take.
As a simple rule of thumb, the best way to approach this question is to solve the two expressions. This gives us the meeting points of the two curves. One of the two meeting points should be the maximum value.
The question is "Find the maximum value of f(x); if f(x) is defined as the Min {(x – 1)^{2} + 2, (x – 2)^{2} + 1}?"
Choice B is the correct answer.
Question 3: Consider functions f(x) = x^{2} + 2x, g(x) = √(x + 1) and h(x) = g(f(x)). What are the domain and range of h(x)?
Answer. Domain: (∞, +∞), Range [0, ∞]
Explanation.
h(x) = g(f(x)) = g(x^{2} + 2x) = √(x^{2} + 2x + 1) = √(x + 1)^{2} = x + 1
This bit is very important, and often overlooked.
√(9) = 3, not ±3
If x^{2} = 9, then x can be ±3, but √(9) is only +3.
So, √(x^{2}) = x, not +x, not ±x
Domain of x + 1 = (∞, +∞), x can take any value.
As far as the range is concerned, x + 1 cannot be negative. So, range = [0, ∞)
The question is "What are the domain and range of h(x)?"
Question 4: [x] = greatest integer less than or equal to x. If x lies between 3 and 5, 5 inclusive, what is the probability that [x^{2}] = [x]^{2}?
A. Roughly 0.64
B. Roughly 0.5
C. Roughly 0.14
D. Roughly 0.36
Answer. Roughly 0.14
Explanation. Let us take a few examples.
[3^{2}] = [3]^{2}
[3.5^{2}] = 12 [3.5]^{2} = 9
[4^{2}] = 16 [4]^{2} = 16
For x ∈ (3, 5). [x]^{2} can only take value 9, 16 and 25.
Let us see when [x^{2}] will be 9, 16 or 25.
If [x^{2}] = 9,
x^{2} ∈ [9, 10)
⇒ x ∈ [3, √10)
[x^{2}] = 16
x^{2} ∈ [16, 17)
⇒ x ∈ [4, √17)
In the given range [x^{2}] = 25 only when x = 5
So [x^{2}] = [x]^{2} when x ∈ [3, √10] or [4, √10) or 5.
The question is "If x lies between 3 and 5, 5 inclusive, what is the probability that [x^{2}] = [x]^{2}?"
Choice C is the correct answer.
Question 5: Give the domain and range of the following functions:
A. f(x) = x^{2} + 1
B. g(x) = log(x + 1)
C. h(x) = 2^{x}
D. f(x) = 1/x+1
E. p(x) = x + 1
F. q(x) = [2x], where [x] gives the greatest integer less than or equal to x
Explanation.
A. f(x) = x^{2} + 1
Domain = All real numbers (x can take any value)
Range [1, ∞). Minimum value of x^{2} is 0.
B. g(x) = log (x + 1)
Domain = Log of a negative number is not defined so (x + 1) > 0 or x > 1
Domain (1, ∞)
Range = (∞, +∞)
Note: Log is one of those beautiful functions that is defined from a restricted domain to all real numbers. Log 0 is also not defined. Log is defined only for positive numbers
C. h(x) = 2^{x} Domain  All real numbers.
Range = (0, ∞)
The exponent function is the mirror image of the log function.
Domain = All real numbers except 1
Range = All real numbers except 0.
E. p(x) = x + 1
Domain = All real numbers
Range = [0, ∞) Modulus cannot be negative
F. q(x) = [2x], where [x] gives the greatest integer less than or equal to x
Domain = All real numbers
Range = All integers
The range is NOT the set of even numbers. [2x] can be odd. [2 * 0.6] = 1. It is very important to think fractions when you are substituting values.
Question 6: How many elements are present in the domain of ^{9–x}C_{x+1}?
A. 5
B. 6
C. 4
D. 7
Answer. 6
Explanation.
For ^{n}C_{r} to be defined, we should have r greater than or equal to zero and, n greater than or equal to r.
Therefore 9 – x ≥ x + 1 or 4 ≥ x and
also x + 1 > 0 or x > –1
Therefore x takes on the values {–1,0,1,2,3,4}
Therefore domain = {–1,0,1,2,3,4} and range = {^{10}C_{0},^{9}C_{1},^{8}C_{2},^{7}C_{3},^{6}C_{4},^{5}C_{5}}
There are 6 elements in the domain.
The question is "How many elements are present in the domain of ^{9–x}C_{x+1}? "
Choice B is the correct answer.
Question 7: f(x + y) = f(x)f(y) for all x, y, f(4) = + 3 what is f(–8)?
A. 1/3
B. 1/9
C. 9
D. 6
Answer. 1/9
Explanation.
f(x + 0) = f(x) f(0)
f(0) = 1
f(4 + – 4) = f(0)
f(4 + – 4) = f(4) f(–4)
1 = +3 * f(–4)
f(4) = 1/3
f(– 8) = f(– 4 + (– 4)) = f(– 4) f(– 4)
The question is "f(x + y) = f(x)f(y) for all x, y, f(4) = + 3 what is f(–8)?"
A. 5
B. 10
C. 6
D. Cannot determine
Answer. 10
Explanation.
f(x – 3) = 2x^{3} + p – qx
Let x = 3, then f(0) = 54 + p – 3q  (1)
f(x^{2} – 4) = x^{2} – 8q + 6p
Let x^{2} = 4, then f(0) = 4 – 8q + 6p  (2)
From (1) and (2)
54 + p – 3q = 4 – 8q + 6p
50 = 5p – 5q
p  q = 10
The question is "If f(x – 3) = 2x^{3 }+ p – qx and f(x^{2 }– 4) = x^{2 }– 8q + 6p, then what is the value of p – q?"
Choice B is the correct answer.
Question 9: 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
A. 1
B. 2
C. 1
D. 3
Answer. 3
Explanation.
f(x) = f(x + 1) + f(x – 1)
Let f(x) = p and f(x – 1) = q
f(x + 1) = f(x + 1 + 1) + f(x + 1 – 1) (Put x = x + 1 here!)
= f(x + 2) + f(x)
p – q = f(x + 2) + p
Or f(x + 2) = q
f(x + 2) = f(x + 2 + 1) + f(x + 2 – 1) (Put x = x + 2 here!)
= f(x + 3) + f(x + 1)
– q = f(x + 3) + p – q
Or f(x + 3) = p
At this point we notice that f(x) = p and f(x+3) = p
f(x) + f(x + 3) = 0 (This is the condition to be satisfied to determine a)
Hence ‘a’ = 3
The question is "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?"
Choice D is the correct answer.
Question 10: x is a real number such that f(x) = 1/x when x > 0 and f(x) = 1/x+1 otherwise. Also f^{n}(x) = f(f^{n  1 }(x)). What is f(3) + f^{2}(3) + f^{3}(3) + f^{4}(3)?
A. 2/3
B. 14/3
C. 0
D. 3
Answer. 14/3
Explanation.
f(x) = 1/x = when x > 0; f(x) = 1/x+1 otherwise
f(3) = 1/3
f^{2}(3) = f(f(3) = f (1/3) = 3
f(3) + f^{2}(3) + f^{3}(3) + f^{4}(3)
= 14/3
The question is "What is f(3) + f^{2}(3) + f^{3}(3) + f^{4}(3)?"
Choice B is the correct answer.
Question 11: Which of the following functions are identical?
g(x) = (√x)^{2}
h(x) = x
A. f(x) and g(x)
B. f(x) and h(x)
C. All 3 are identical
D. None of these
Answer. None of these
Explanation.
For functions to be identical, their domains should be equal
f(x) – x can’t be zero
g(x) – x can’t be negative
h(x) – x can take all possible values
The question is "Which of the given functions are identical?"
Choice D is the correct answer.
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