Tag: introduction to sets

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Multiple choice business maths functions and graphs some functions and their graphs -i graphs of the form y=ax^2+bx+c introduction to sets

Time complexity to check if an edge exists between two vertices would be __________.

  1. O(V*V)

  2. O(V+E)

  3. O(1)

  4. O(E)
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

In an adjacency list representation, checking if an edge exists between two vertices u and v requires iterating through the list of neighbors of u, which takes O(degree(u)) time. In the worst case, this is O(E) or O(V). O(E) is the most appropriate choice among the options.

Multiple choice business maths functions and graphs some functions and their graphs -i graphs of the form y=ax^2+bx+c introduction to sets

Which of the following functions are identity functions?

  1. $f:R\rightarrow R, f(x) = x$
  2. $g : N \rightarrow Z, g(p)= 3$
  3. $h:z \rightarrow z, h(y)=y$
  4. $g:N\rightarrow N, g(z) =z$
Reveal answer Fill a bubble to check yourself
A,C,D Correct answer
Explanation

An identity function is a function that always returns the same value that was used as its argument.

Hence $f:R\rightarrow R,f(x)=x$ is an identity function

Multiple choice business maths functions and graphs some functions and their graphs -i graphs of the form y=ax^2+bx+c introduction to sets

If ${ (x, 2), (4, y) }$ represents an identity function, then $( x, y)$ is :

  1. (2, 4)

  2. (4, 2)

  3. (2, 2)

  4. (4, 4)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Identity funtion returns the same value as of input.

Since input $x$ gives output $2$  
$\therefore$ $x=2$
Similarly input of $4$ gives output $y$
$\therefore y=4$
$\Rightarrow (x,y)=(2,4)$

Option (a) is correct.

Multiple choice business maths functions and graphs some functions and their graphs -i graphs of the form y=ax^2+bx+c introduction to sets

An identity function is a?

  1. Many to many function

  2. One to One function

  3. Many to one function

  4. None
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

An identity function is of the form $f(x) = cx$


A one-to-one function is a function that preserves distinctness; it never maps distinct elements of its domain to the same element of its codomain.

In other words, every element of the function's codomain is the image of at most one element of its domain.

As $f(x) =cx$ has a different value at every value of $x$, it is a One to One function.

Multiple choice business maths functions and graphs some functions and their graphs -i graphs of the form y=ax^2+bx+c introduction to sets

State whether the following statement is True or False.
The inverse of an identity function is the identity function itself.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

An identity function $f(x)$ is of the form $f(x) = cx$

Let inverse of $f(x)$ be $g(x)$
$\therefore g(x) = \dfrac{x}{c}$ (Identity Function)
Thus inverse of an identity function is the identity function itself.

Multiple choice business maths functions and graphs some functions and their graphs -i graphs of the form y=ax^2+bx+c introduction to sets

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function such that for any irrational number $r,$ and any real number $x$ we have $f(x)=f(x+r)$. Then, $f$ is

  1. an identity function

  2. a constant function

  3. a zero function

  4. onto function
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A constant function is a function that has the same output value no matter what your input value is. Because of this, a constant function has the form y=k where k is a constant(a single value that doesn't change).

So in here our input is $x$ i f we change our input to say $x+r$
and still output does'nt change i.e.
$f(x)=f(x+r)$
it means it is a constant function,
example:
$ f(x)=5=5(x)^{0} $
$ f(x+r)=5(x+r)^{0}$
$ f(x+r)=5\times 1 $
$ f(x+r)=5=f(x) $

Multiple choice business maths functions and graphs some functions and their graphs -i graphs of the form y=ax^2+bx+c introduction to sets

Let $f$ be a linear function for which $f (6)  - f (2) = 12$. The value of $f (12) - f(2)$ is equal the 

  1. $12$
  2. $18$
  3. $24$
  4. $30$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let $f(x)=ax+b$ [Since $f$ is given to be linear function] where $a$ and $b$ are constants.

According to the problem 
$f(6)-f(2)=12$
or, $6a-b-2a-b=12$
or, $4a=12$
or, $a=3$.
Now 
$f(12)-f(2)$
$=12a+b-2a-b$
$=10a=30$. [Using value of $a$]