Relations and Functions — Practice

Question 1

If A = {1, 2} and B = {a, b, c}, find A×B and state n(A×B).

Accepted Answer

A×B = {(1,a),(1,b),(1,c),(2,a),(2,b),(2,c)}. n(A×B) = n(A)×n(B) = 2×3 = 6.

Question 2

Find the domain and range of f(x) = √(9 − x²).

Accepted Answer

Step 1: Require 9−x² ≥ 0, so x² ≤ 9, giving −3 ≤ x ≤ 3. Domain = [−3, 3]. Step 2: At x=0, f(0)=3 (maximum). At x=±3, f=0 (minimum). f is non-negative throughout. Range = [0, 3].

Question 3

Let f: R→R be defined by f(x) = 2x+3. Prove that f is bijective.

Accepted Answer

One-one: Suppose f(x₁)=f(x₂). Then 2x₁+3 = 2x₂+3, so 2x₁=2x₂, giving x₁=x₂. Hence f is one-one. Onto: Let y∈R be arbitrary. Set x=(y−3)/2. Then x∈R and f(x) = 2·(y−3)/2+3 = y−3+3 = y. So every y has a pre-image. Hence f is onto. Since f is both one-one and onto, f is bijective.

Practice — Relations and Functions

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