A function f: ℝ → ℝ maps each real number in the domain ℝ to a real number in the range ℝ. For each input x ∈ ℝ, there is a corresponding output f(x) ∈ ℝ.
Class 12 Maths Relations and Functions Chapter 1 for CBSE Exam 2024-25 introduces relations between sets and various types of functions like one-one and onto. It covers domain and range of functions composite functions and inverse functions. This chapter is essential for understanding concepts used in higher mathematics and further studies.
Share
The function f: ℝ → ℝ defined by f(x) = x³ is a function from the real numbers to the real numbers.
Analyzing the function:
1. One-to-one (Injective):
A function is one-to-one if different inputs map to different outputs.
Suppose f(x₁) = f(x₂), i.e., x₁³ = x₂³.
Then, x₁ = x₂.
This demonstrates that f(x) = x³ is one-to-one (injective).
2. Onto (Surjective): A function is onto if every element in the target set, here ℝ, has a pre-image in the domain. For any real number y ∈ ℝ, we can find an x ∈ ℝ such that f(x) = x³ = y.
Specifically, x = ∛y. This demonstrates that the function f(x) = x³ is onto (surjective).
Conclusion: The function f(x) = x³ is one-to-one and onto, so the correct answer is: – One-one and onto.
Click for more:
https://www.tiwariacademy.com/ncert-solutions/class-12/maths/#chapter-1