A bell shaped membership function is specified by three parameters (𝑎,𝑏,𝑐)…

Answer: μ(x) = 1 / (1 + ((x - c)/a)^(2b))

Key properties of this form:

• Peak at x = c: when x = c the powered term is 0, so μ(c) = 1/(1+0) = 1.

• Tails go to zero: as |x - c| → ∞ the powered term → ∞, so μ(x) → 0, producing bell-shaped tails.

• Even exponent ensures symmetry and nonnegativity: using (2b) makes ((x - c)/a)^(2b) ≥ 0 for all x, so the curve is symmetric about c and well defined for real-valued b.

• Parameter interpretations: c is the center (location of peak), a scales the width (larger a → wider bell), and b controls steepness (larger b → steeper sides).

Why the other presented expressions are not appropriate:

• Expression with exponent b but without squaring (1/(1 + ((x - c)/a)^b)): may not be symmetric or well defined for non-integer b because the base ((x - c)/a) can be negative; this can break the bell-shape property.

• Expressions without the reciprocal (1 + ((x - c)/a)^b or 1 + ((x - c)/a)^(2b)) are not normalized into [0,1]; they produce values ≥ 1 (or possibly negative), so they are not valid membership degrees.

Summary: The correct bell-shaped membership function is μ(x) = 1 / (1 + ((x - c)/a)^(2b)) because it is centered at c, bounded between 0 and 1, symmetric, and its parameters have clear roles controlling center, width, and steepness.