Let f(x) = x², in R, then the range of f will be:

Concept:

Range of a Function:

• The range of a function is the set of all possible output values (f(x)) for the given domain.
• For the function f(x) = x², where x ∈ ℝ, the output is always a non-negative real number.
• This is because the square of any real number is always greater than or equal to zero.
• Minimum value of f(x) occurs at x = 0, which gives f(0) = 0.
• As x increases or decreases, f(x) increases without any upper limit.

Calculation:

Given,

Function: f(x) = x²

Domain: x ∈ ℝ

⇒ f(0) = 0 ∈ range

⇒ For x = ±1, ±2, f(x) = 1, 4, which are > 0

⇒ f(x) ≥ 0 for all real x

⇒ So, the range is [0, ∞)

⇒ f(x) can take any value from 0 to ∞

The range of f(x) is the set of all non-negative real numbers.

Understanding Positive vs Non-negative Real Numbers:

• Positive real numbers are all real numbers greater than zero, i.e., (0, ∞), and do not include 0.
• Non-negative real numbers include all positive real numbers and 0, i.e., [0, ∞).
• The function f(x) = x² is defined for all real numbers, and it gives an output ≥ 0 for any x ∈ ℝ.
• At x = 0, f(x) = 0² = 0, which is part of the range.
• Since 0 is not included in positive real numbers, option 3 is incorrect.
•  

∴ Option 3 is incorrect because it excludes 0, which is part of the range.

Hence Option 1 is the correct answer