Which of the following is a one-one (injective) function from ℝ to ℝ?

Solution

Maths sir · 3 steps

Hunt for two different inputs with the same output; if you can find them, the function is not one-one.

1. What one-one means. A function is one-one if different inputs always give different outputs. So f(x₁) = f(x₂) must force x₁ = x₂.

   Function	Two inputs, same output?	One-one?
   f(x) = x²	f(-2) = f(2) = 4	No
   f(x) = x³	None: x₁³ = x₂³ gives x₁ = x₂	Yes
   f(x) = |x|	f(-3) = f(3) = 3	No
   f(x) = sin x	sin 0 = sin π = 0	No

2. Why the cube works. x₁³ - x₂³ = (x₁ - x₂)(x₁² + x₁ x₂ + x₂²). The second bracket is zero only when x₁ = x₂ = 0, so the product is zero only when x₁ = x₂. Also, x³ is strictly increasing on ℝ.
3. So. Only f(x) = x³ is one-one from ℝ to ℝ, which is option (b).

The idea to keep. To show a function is not one-one, one pair of inputs sharing an output is enough. Even functions and periodic functions are never one-one on all of ℝ.

Answer from the NTA final answer key. Solution by PrepWiser.