Relations and Functions — Chapter Test | Class 11 IMO

Q1

The vertical line test is used to determine whether:

The vertical line test checks if any vertical line intersects a graph more than once. If it does, the relation assigns multiple y-values to one x-value, and thus is not a function.

Q2

If f(x)=5x and g(x)=2x, then (f−g)(x) equals:

5x−2x=3x.

Q3

If f(x)=2x+1, then f(4) is:

2×4+1=9.

Q4

If f(x) = |x| / x for x ≠ 0, and f(0) = 0, this function is known as the:

The definition f(x) = |x|/x for x ≠ 0 and 0 for x=0 is the standard definition of the Signum Function.

Q5

The number of relations from a set having 2 elements to itself is:

A×A has 4 elements, so relations = 2⁴ = 16.

Q6

An express train from Chennai to Madurai travels at a uniform speed of 80 km/h. The distance d(t) covered in t hours is a function. What is its domain if the trip takes 6 hours?

The time t starts from 0 hours and ends at 6 hours. Therefore, the domain of the function d(t) is [0, 6].

Q7

Anita maps the age of her classmates to their heights. Can this relation be considered a function?

If 'Age' is the input, a single age (like 16) might correspond to multiple students with different heights. One input giving multiple outputs means it's not a function.

Q8

In a school assembly in Mumbai, 5 students are to sit in 5 designated chairs. If each chair can be occupied by exactly one student, what type of mapping does this represent from students to chairs?

Since each of the 5 students sits in exactly one distinct chair and all 5 chairs are occupied, it is a One-One and Onto mapping (Bijective).

Q9

A function which is not onto is called:

Into functions do not cover entire codomain.

Q10

Which of the following is true for the Cartesian product of empty sets?

By definition, the Cartesian product of any set with an empty set is the empty set. There are no ordered pairs to form.

Q11

Two sets are given: A = {a, b} and B = {c, d}. Is A × B equal to B × A?

The Cartesian product is generally non-commutative. A × B ≠ B × A unless the sets A and B are identical.

Q12

If f(x) = x³ and g(x) = ∛x, what is f(g(x))?

f(g(x)) = (∛x)³ = x. They are inverse functions of each other.

Q13

Which of the following functions is neither even nor odd? f(x) = ?

An even function has f(−x) = f(x). An odd function has f(−x) = −f(x). For f(x) = x² + x, f(−x) = x² − x, which is neither f(x) nor −f(x).

Q14

For f(x) = √x, which of the following is true regarding its domain and range?

Square roots of negative numbers are not real, so x ≥ 0 (Domain is [0, ∞)). The principal square root is always non-negative, so Range is [0, ∞).

Q15

If A × B = {(a, x), (a, y), (b, x), (b, y)}, find the set A.

Set A is the set of all first elements of the ordered pairs in A × B. Thus, A = {a, b}.