Master Types of Functions for CBSE Class 11 Applied Mathematics. Learn One-One, Onto, Constant, and Identity functions with solved examples and exercises.
A function f: A → B is one-one (injective) if distinct elements of A have distinct images in B. If f(x₁) = f(x₂) implies x₁ = x₂, the function is one-one. Otherwise, it is many-one.
Step 1: Let f(x₁) = f(x₂).
Step 2: 2x₁ + 3 = 2x₂ + 3.
Step 3: Subtract 3 from both sides: 2x₁ = 2x₂.
Step 4: Divide by 2: x₁ = x₂.
Answer: Since x₁ = x₂, the function is one-one.
A function f: A → B is onto (surjective) if every element in the codomain B has at least one pre-image in the domain A. If there exists an element in B with no pre-image, it is into.
Step 1: The codomain is R (all real numbers).
Step 2: The range of x² is [0, ∞).
Step 3: Since [0, ∞) ≠ R, there are negative numbers with no pre-image.
Answer: The function is not onto.
A constant function maps every element of the domain to a single fixed value in the codomain. An identity function maps every element to itself.
Step 1: Identify the function type: constant.
Step 2: For any input x, f(x) = 7.
Step 3: Substitute x = 5.
Answer: f(5) = 7.