Functions Question and Solution

Question

Consider set A with 'a' elements, set B with 'b' elements, set C with 'c' elements. We can define a function that is one-one but not onto from set A to set B, a function that is onto but not one-one from set B to set C; and a function that is injective but not surjective from C to A. Arrange a, b, c in ascending order.

Explanation

CAT Functions, Onto functions

  Question

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

Answer: Choice (C)

Explanatory Answer:

First let us think of the number of potential functions possible. Each element in A has three options in the co-domain. So, the number of possible functions = 3⁴ = 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 co-domain without a pre-image).

Ø  Let us assume that elements are mapped into A and B. Number of ways in which this can be done = 2⁴ – 2 = 14

o   2⁴ because the number of options for each element is 2. Each can be mapped on to either A or B

o   -2 because these 2⁴ 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

Ø  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 pre-image). 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 co-doamin remains without a pre-image - Number of functions where 2 elements from the co-doamin remain without a pre-image

ð  Total number of onto functions = 81 – 42 – 3 = 81 – 45 = 36  

Answer Choice (C)

CAT Functions

Question

Qn. 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 Choice: B

Explanatory Answer

f(x + 0) = f(x) f(0)

f(0) = 1

f(4 + -4) = f(0)

f(4 + -4) = f(4) f(-4)

1 = +3 x f(-4)

f(-4) = 1/3

f(-8) = f(-4 + (-4)) = f(-4) f(-4)

f(-8) = 1/3 x 1/3 = 1/9 

Answer Choice (B)

CAT Functions

Questions from functions have increased over the past few years. Given below is an interesting question from functions.

Question

Consider functions  f(x) = x² + 2x,  g(x) = √(x +1) and h(x) = g(f(x)). What are the domain and range of h(x)?

Correct Answers:
Domain = ( - infinity, +infinity)
Range - [0, infinity] 

Explanatory Answer

h(x) = g(f(x)) = g(x² + 2x) = Ö( x² + 2x + 1) = Ö(x+1)² = | x + 1|

This bit is very important, and often overlooked

sqrt(9) = 3, not +3

If x² = 9, then x can be +3, but Sqrt(9) is only +3.

So, Sqrt(x²)= |x|, not +x, not + x

Domain of | x + 1| = ( -infinity, + infinity), x can take any value.

As far as the range is concerned, | x + 1| cannot be negative. So, range = [0, infinity)

CAT Functions - Questions

Have given below a few questions on functions.

1. Give the domain and range of the following functions

i. f(x) = x² + 1
ii. g(x) = log (x + 1)
iii. h(x) = 2^x
iv. f(x) = 1/ (x+1)
v. p(x) = | x + 1|
vi. q(x) = [2x], where [x] gives the greatest integer less than or equal to x

Happy cracking