ⵜⴰⵙⵖⵏⵜ (ⵜⵓⵙⵏⴰⴽⵜ)

ⵜⴰⵙⵖⵏⵜ (ⵜⵓⵙⵏⴰⴽⵜ)
mathematical concept
ⴰⴷⵓⵙⵎⵉⵍ ⵏ	binary relation, partial function
ⵜⴰⵥⵍⴰⵢⵜ ⵏⵏⴰ ⵙⵉⵙ ⵉⵏⵏⵣⴳⵎⵏ	ⵜⵓⵙⵏⴰⴽⵜ
ⵉⵍⵍⴰ ⴳ ⵓⵙⵖⵓⵏ	https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:functions, https://www.khanacademy.org/math/linear-algebra/matrix-transformations/linear-transformations/v/a-more-formal-understanding-of-functions
ⴰⵣⵎⵎⴻⵎ	maplet
ⴰⵏⵎⴳⴰⵍ	multivalued function
ⵜⴰⵍⴳⴰⵎⵜ ⵏ ⵓⴼⵔⴷⵉⵙ ⴰ	list of mathematical functions
	ⵙⵏⴼⵍ ⴰⵙⴰⴳⵎ - ⵙⵏⴼⵍ - ⵡⵉⴽⵉⴷⴰⵜⴰ

ⴰⵎⴰⴳⵔⴰⴷ ⴰⴷ ⵉⴳⴰ ⵙⵓⵍ ⴰⴳⵓⵊⵉⵍ. ⵎⴽ ⵜⵓⴼⵉⴷ, ⵔⵏⵓ ⴰⵙⵖⵓⵏ ⵏⵏⵙ ⴳ ⴽⵕⴰⴹⵜ ⵜⴰⵙⵏⵉⵡⵉⵏ ⵢⴰⴹⵏⵉⵏ.

ⵜⴰⵙⵖⵏⵜ (ⵙ ⵜⵏⴳⵍⵉⵣⵜ: function), ⴳ ⵜⵓⵙⵏⴰⴽⵜ, ⵜⴳⴰ ⵢⴰⵏ ⵓⵍⵓⴳⵏ ⵉⵣⴷⴰⵢⵏ ⴽⵓ ⴰⴼⵔⴷⵉⵙ ⵙⴳ ⵜⴳⵔⵓⵎⵎⴰ ⵜⴰⵎⵣⵡⴰⵔⵓⵜ, ⵉⵜⵜⵓⵙⵎⵎⴰⵏ ⵜⴰⵖⵓⵍⵜ, ⵙ ⵢⴰⵏ ⵓⴼⵔⴷⵉⵙ ⴰⵎⵢⵉⵡⵏ ⵙⴳ ⵜⴳⵔⵓⵎⵎⴰ ⵜⵉⵙⵙ ⵙⵏⴰⵜ, ⵉⵜⵜⵓⵙⵎⵎⴰⵏ ⵜⴰⵣⴷⴰⵖⵓⵍⵜ. ⵉⴳⴰ ⵓⵔⵎⵎⵓⵙ ⵏ ⵜⵙⵖⵏⵜ ⴰⵔⵙⵍⴰⵏ ⴳ ⵓⵎⴰⵜⴰ ⵏ ⵉⴳⵔⴰⵏ ⵏ ⵜⵓⵙⵏⴰⴽⵜ ⴷ ⵜⵓⵙⵙⵏⵉⵡⵉⵏ ⵢⴰⴹⵏⵉⵏ. ⵜⵓⵇⵇⵏⴰ ⴰⴷ ⴷⴰ ⵜⴻⵜⵜⵓⵙⴽⴰⵏ ⴳ ⵓⵎⴰⵜⴰ ⵣⵓⵏ ⴷ "ⵜⴰⵎⴰⵛⵉⵏⵜ" ⵉⵜⵜⴰⵎⵥⵏ ⵜⵉⵏⴽⵛⵓⵎⵉⵏ (ⴰⴼⵔⴷⵉⵙ ⵙⴳ ⵜⴰⵖⵓⵍⵜ) ⵜⴼⴽ ⵜⵓⴼⴼⵖⴰ ⵜⴰⵎⵢⵉⵡⵏⵜ (ⴰⴼⵔⴷⵉⵙ ⵏ ⵜⵣⴷⴰⵖⵓⵍⵜ). ⵙ ⵓⵎⴷⵢⴰ, ⵜⴰⵙⵖⵏⵜ ⵍⵍⵉ ⵉⵣⴷⴰⵢⵏ ⴽⵓ ⴰⵎⴹⴰⵏ ⴰⵏⵉⵍⴰⵡ $x$ ⵙ ⵓⵎⴽⴽⵓⵥ ⵏⵏⵙ $x^{2}$ ⵜⴳⴰ ⵜⴰⵙⵖⵏⵜ.^[1]

ⴰⵙⵏⵎⵍ ⴰⵍⵖⴰⵏ

ⵉⴱⴷⴷ ⵓⵙⵏⵎⵍ ⴰⵍⵖⴰⵏ ⵏ ⵜⵙⵖⵏⵜ ⵖⴼ ⵓⵔⵎⵎⵓⵙ ⵏ ⵜⵢⵓⴳⵉⵡⵉⵏ ⵉⵏⵎⴰⵍⴰⵏ. ⵜⴰⵙⵖⵏⵜ $f$ ⵙⴳ ⵜⴳⵔⵓⵎⵎⴰ $X$ (ⵜⴰⵖⵓⵍⵜ) ⵖⵔ ⵜⴳⵔⵓⵎⵎⴰ $Y$ (ⵜⴰⵣⴷⴰⵖⵓⵍⵜ) ⵜⴳⴰ ⵢⴰⵜ ⵜⴳⵔⵓⵎⵎⴰ ⵏ ⵜⵢⵓⴳⵉⵡⵉⵏ ⵉⵏⵎⴰⵍⴰⵏ $(x,y)$ ⵍⵍⵉ ⴳ ⵉⴳⴰ $x$ ⴰⴼⵔⴷⵉⵙ ⵙⴳ $X$ ⴷ $y$ ⵉⴳⴰ ⴰⴼⵔⴷⵉⵙ ⵙⴳ $Y$ , ⵙ ⵜⴼⴰⴷⴰ ⵏ ⴰⴷ ⵢⵓⵎⴰⵏ ⴽⵓ ⴰⴼⵔⴷⵉⵙ $x$ ⴳ $X$ ⵢⴰⵜ ⵜⴽⴽⵍⵜ ⵀⵍⵍⵉ ⴰⵎⵎ ⵉⵙⴳⵔ ⴰⵎⵣⵡⴰⵔⵓ ⴳ ⵜⵢⵓⴳⴰ. ⴷⴰ ⵜⴻⵜⵜⵢⴰⵔⴰ ⵜⵓⵇⵇⵏⴰ ⴰⴷ ⵙ ⵉⵎⴽⴰ $f:X\to Y$ . ⵉⴳ ⵉⵍⵍⴰ $(x,y)$ ⴳ ⵜⵙⵖⵏⵜ $f$ , ⴷⴰ ⵏⵜⵜⵉⵏⵉ ⵎⴰⵙ ⴷ $y$ ⵜⴳⴰ ⵜⵓⴳⵏⴰ ⵏ $x$ ⵙⴳ $f$ , ⵏⴰⵔⴰ $y=f(x)$ .^[2]

ⵜⴻⵜⵜⵓⵙⵎⵎⴰ ⵜⴳⵔⵓⵎⵎⴰ ⵏ ⵡⴰⵣⴰⵍⵏ ⴰⴽⴽⵯ $f(x)$ ⵍⵍⵉ ⵏⵥⴹⴰⵕ ⴰⴷ ⵏⴰⴼ ⵜⵓⴳⵏⴰ ⵏ ⵜⵙⵖⵏⵜ $f$ ⵏⵖ ⴰⵣⵉⵍⴰⵍ ⵏⵏⵙ. ⵉⵇⵇⴰⵏ ⴷ ⴰⴷ ⵓⵔ ⵏⵙⵎⵔⴽⴰⵙ ⴳⵔ ⵜⵓⴳⵏⴰ (ⴰⵣⵉⵍⴰⵍ) ⴷ ⵜⵣⴷⴰⵖⵓⵍⵜ. ⵜⴰⵣⴷⴰⵖⵓⵍⵜ ⵜⴳⴰ ⵜⴰⴳⵔⵓⵎⵎⴰ $Y$ ⵍⵍⵉ ⴳ ⵥⴹⴰⵕⵏ ⴰⴷ ⵉⵍⵉⵏ ⵡⴰⵣⴰⵍⵏ ⵏ ⵜⵓⴼⴼⵖⴰ, ⵎⴰⵛⴰ ⵜⵓⴳⵏⴰ (ⴰⵣⵉⵍⴰⵍ) ⵜⴳⴰ ⵜⴰⴳⵔⵓⵎⵎⴰ ⵏ ⵡⴰⵣⴰⵍⵏ ⵏ ⵜⵓⴼⴼⵖⴰ ⵍⵍⵉ ⵏⵉⵜ ⵉⴳⴰⵏ ⵜⴰⵢⴰⴼⵓⵜ. ⵙ ⵓⵎⴷⵢⴰ, ⴳ ⵜⵙⵖⵏⵜ $f(x)=x^{2}$ , ⵉⴳ ⵜⴳⴰ ⵜⴰⵖⵓⵍⵜ ⴷ ⵜⵣⴷⴰⵖⵓⵍⵜ ⵜⴳⵔⵓⵎⵎⴰ ⵏ ⵉⵎⴹⴰⵏⵏ ⵉⵏⵉⵍⴰⵡⵏ $\mathbb {R}$ , ⵀⴰⵜ ⴰⵣⵉⵍⴰⵍ (ⵜⵓⴳⵏⴰ) ⵉⴳⴰ ⵜⴰⴳⵔⵓⵎⵎⴰ ⵏ ⵉⵎⴹⴰⵏⵏ ⵉⵏⵉⵍⴰⵡⵏ ⵓⵎⵏⵉⴳⵏ $[0,+\infty )$ , ⵍⵍⵉ ⵉⴳⴰⵏ ⵖⴰⵙ ⴰⴳⵣⵣⵓⵎ ⵙⴳ ⵜⵣⴷⴰⵖⵓⵍⵜ.^[3]

ⵜⵉⵔⵔⴰ

ⵍⵍⴰⵏⵜ ⵎⵏⵏⴰⵡⵜ ⵜⵖⴰⵔⴰⵙⵉⵏ ⵏ ⵜⵉⵔⵔⴰ ⵏ ⵜⵙⵖⵏⵉⵏ. ⵜⴰⵖⴰⵔⴰⵙⵜ ⴰⴽⴽⵯ ⵉⵜⵜⵓⵙⵙⴰⵏⵏ ⴳⴰⵏⵜ ⵜⵜ ⵜⵉⵔⵔⴰ ⵜⵉⵙⵖⵏⴰⵏⵉⵏ, ⵣⵓⵏ ⴷ f(x), ⵍⵍⵉ ⵉⵙⵙⵎⵔⵙ ⵓⵎⵙⵏⴰⴽⵜ ⵍⵢⵓⵏⴰⵔⴷ ⵓⵍⵔ ⴳ ⵓⵙⴳⴳⵯⴰⵙ ⵏ 1734. ⵖⵉⴷ, f ⵉⴳⴰ ⵉⵙⵎ ⵏ ⵜⵙⵖⵏⵜ, ⴷ x ⵉⴳⴰ ⴰⵎⵙⴽⵉⵍ ⴰⵙⵉⵎⴰⵏ ⵉⵜⵜⴳⵏⵙⴻⵙⵏ ⵢⴰⵜ ⵜⵏⴽⵛⵓⵎⵜ.^[4]

ⴷⴰ ⵉⵙⵓⵜⵜⵓ ⵓⵙⵏⴳⵍ ⵏ ⵓⵍⴳⴳⵉ ⵜⴰⵖⵓⵍⵜ ⴷ ⵜⵣⴷⴰⵖⵓⵍⵜ ⵏⵏⵉⵛⴰⵏ. ⴰⴷ ⵏⴰⵔⴰ f: X → Y, ⴰⵏⴰⵎⴽ ⵏⵏⵙ ⵎⴰⵙ ⴷ f ⵜⴳⴰ ⵜⴰⵙⵖⵏⵜ ⵉⵣⴷⴰⵢⵏ ⵉⴼⵔⴷⵉⵙⵏ ⵙⴳ ⵜⴳⵔⵓⵎⵎⴰ X ⵖⵔ ⵜⴳⵔⵓⵎⵎⴰ Y. ⵉⵜⵜⵓⵙⴽⴰⵏ ⵓⵍⵓⴳⵏ ⵢⵓⵜⵜⴰⵏ ⵉⵣⴷⴰⵢⵏ ⴰⵏⴽⵛⵓⵎ x ⴰⴽⴷ ⵜⵓⴼⴼⵖⴰ y ⵙ ⵓⵍⴳⴳⵉ ⵢⴰⴹⵏⵉⵏ: x ↦ y. ⵙ ⵓⵎⴷⵢⴰ, ⵜⵖⵉⵢ ⴰⴷ ⵜⴻⵜⵜⵢⴰⵔⴰ ⵜⵙⵖⵏⵜ ⵏ ⵓⵙⴽⴽⵓⵥ ⵙⴳ ⵉⵎⴹⴰⵏⵏ ⵉⵏⵉⵍⴰⵡⵏ ⵖⵔ ⵉⵎⴹⴰⵏⵏ ⵉⵏⵉⵍⴰⵡⵏ ⵙ: f: ℝ → ℝ, ⵍⵍⵉ ⴳ x ↦ x².^[5]

ⵜⴰⴳⵏⵙⴻⵙⵜ ⵏ ⵜⵙⵖⵏⵉⵏ

ⵥⴹⴰⵕⵏⵜ ⴰⴷ ⵜⵜⵓⴳⵏⵙⴻⵙⵏⵜ ⵜⵙⵖⵏⵉⵏ ⵙ ⵎⵏⵏⴰⵡⵜ ⵜⵖⴰⵔⴰⵙⵉⵏ:

ⵙ ⵜⵏⴼⴰⵍⵉⵜ ⵜⴰⵎⵚⵍⴰⴹⵜ

ⵜⴰⵖⴰⵔⴰⵙⵜ ⴰⴽⴽⵯ ⵉⴷⵓⵙⵏ ⵉ ⵜⴳⵏⵙⴻⵙⵜ ⵏ ⵜⵙⵖⵏⵜ ⵉⴳⴰ ⵜⵜ ⵓⵙⵏⴼⴰⵍⵉ ⴰⵎⵚⵍⴰⴹ (ⵏⵖ ⵜⴰⵏⴼⴰⵍⵉⵜ), ⵍⵍⵉ ⵉⴳⴰⵏ ⴰⵙⴷⴷⵉ ⵏ ⵜⵎⵀⵍⵉⵏ ⵜⵓⵙⵏⴰⴽⵜⴰⵏⵉⵏ ⵖⴼ ⵓⵎⵙⴽⵉⵍ x. ⵙ ⵓⵎⴷⵢⴰ: f(x) = 3x² - 4x + 2 ⴷⴰ ⵜⴻⵜⵜⴳⵍⴰⵎ ⵜⵏⴼⴰⵍⵉⵜ ⴰⴷ ⴰⵍⵓⴳⵏ ⵉ ⵓⵙⵉⴹⵏ ⵏ ⵜⵓⴼⴼⵖⴰ ⵉ ⴽⵔⴰⵢⴳⴰⵜⵜ ⵜⴰⵏⴽⵛⵓⵎⵜ x ⵉⵜⵜⵓⴼⴽⴰⵏ. ⵜⵜⵓⵙⵎⵎⴰⵏⵜ ⵜⵙⵖⵏⵉⵏ ⴰⴷ ⵜⵉⵙⵖⵏⵉⵏ ⵜⵉⴳⵜⴼⵓⵍⵉⵏ. ⵣⴹⴰⵕⵏⵜ ⵜⵙⵖⵏⵉⵏ ⵢⴰⴹⵏⵉⵏ ⴰⴷ ⴰⵎⵓⵏⵜ ⵜⵉⵎⵀⵍⵉⵏ ⵣⵓⵏ ⴷ ⵜⵉⴽⵕⴷⵉⵙⴽⵜⴰⵏⵉⵏ (sin(x), cos(x)), ⵜⵉⵙⵏⵏⴼⵙⴰⵔⵉⵏ (eˣ), ⵏⵖ ⵜⵉⵍⵓⴳⴰⵔⵉⵜⵎⵉⵢⵉⵏ (log(x)).^[6]

ⵙ ⵓⵎⵙⴽⴰⵏ

ⴰⵎⵙⴽⴰⵏ ⵏ ⵜⵙⵖⵏⵜ f ⵉⴳⴰ ⵜⴳⵔⵓⵎⵎⴰ ⵏ ⵜⵢⵓⴳⵉⵡⵉⵏ ⵉⵏⵎⴰⵍⴰⵏ (x, f(x)) ⵍⵍⵉ ⴳ ⵉⵍⵍⴰ x ⴳ ⵜⴰⵖⵓⵍⵜ ⵏ f. ⴷⴰ ⵜⵜⵓⴳⵏⵙⴻⵙⵏⵜ ⵜⵢⵓⴳⵉⵡⵉⵏ ⴰⴷ ⵙ ⵜⵇⵉⴼⴰ ⴳ ⵓⵏⴳⵔⴰⵡ ⵏ ⵉⵎⵙⵉⴷⴰⴳⵏ ⵉⴽⴰⵔⵜⵉⵣⵢⴰⵏⵏ, ⵍⵍⵉ ⴳ ⴷⴰ ⵉⵜⵜⴳⵏⵙⴻⵙ ⵓⴳⵍⵍⵓⵙ ⴰⵎⵏⵉⴷⴰⵏ ⴰⵎⵙⴽⵉⵍ x ⴷ ⵓⴳⵍⵍⵓⵙ ⴰⴱⴷⴷⴰⵢ ⴷⴰ ⵉⵜⵜⴳⵏⵙⴻⵙ ⴰⵎⵙⴽⵉⵍ y = f(x). ⴷⴰ ⵢⴰⴽⴽⴰ ⵓⵎⵙⴽⴰⵏ ⴰⵙⴽⵙⵉⵡ ⴰⵎⵥⵕⴰⵡ ⴰⵎⴰⵜⴰⵢ ⵖⴼ ⵓⵎⵓⴳⴳⵉ ⵏ ⵜⵙⵖⵏⵜ, ⵣⵓⵏ ⴷ ⵎⴰⵏⵉ ⴳ ⴷⴰ ⵜⴻⵜⵜⵏⵔⵏⴰⵢ ⵏⵖ ⴷⴰ ⵜⴻⵜⵜⵏⵓⵙⵔⵓ. ⴰⵔⴰⵎ ⵏ ⵓⵣⵔⵉⴳ ⴰⴱⴷⴷⴰⵢ ⵉⴳⴰ ⵢⴰⵜ ⵜⵖⴰⵔⴰⵙⵜ ⵏ ⵓⵙⵙⵉⴷⵜ ⵏ ⵉⵙ ⴷⴰ ⵉⵜⵜⴳⵏⵙⴻⵙ ⵢⴰⵏ ⵓⵎⴽⵏⴰⵡ ⵜⴰⵙⵖⵏⵜ: ⵉⴳ ⵉⵎⵢⴰⴱⴰⵢ ⴽⵔⴰ ⵏ ⵓⵣⵔⵉⴳ ⴰⴱⴷⴷⴰⵢ ⴷ ⵓⵎⵙⴽⴰⵏ ⴳ ⵓⴳⴳⴰⵔ ⵏ ⵢⴰⵜ ⵜⵇⵉⴼⵉⵜ, ⵉⵍⵎⵎⴰ ⴰⵎⴽⵏⴰⵡ ⵓⵔ ⵉⴳⵉ ⴰⵎⵙⴽⴰⵏ ⵏ ⵜⵙⵖⵏⵜ.^[7]

ⵙ ⵜⴼⵍⵡⵉⵜ ⵏ ⵡⴰⵣⴰⵍⵏ

ⴳ ⵜⵙⵖⵏⵉⵏ ⴷⴰⵔ ⵜⵍⵍⴰ ⵜⴰⵖⵓⵍⵜ ⵢⵓⵜⵜⴰⵏ, ⵏⵥⴹⴰⵕ ⴰⴷ ⵜⵏⵜ ⵏⴳⵙⴻⵙ ⵙ ⵓⵙⵎⵔⵙ ⵏ ⵜⴼⵍⵡⵉⵜ. ⴷⴰ ⵜⵎⵎⴰⵍ ⵜⴼⵍⵡⵉⵜ ⴰⴷ ⵉ ⴽⵓ ⴰⵏⴽⵛⵓⵎ ⴰⵣⴰⵍ ⵏ ⵜⵓⴼⴼⵖⴰ ⵏⵏⵙ. ⵙ ⵓⵎⴷⵢⴰ, ⵉⴳ ⵜⴳⴰ ⵜⴰⵖⵓⵍⵜ {1, 2, 3, 4} ⴷ ⵓⵍⵓⴳⵏ ⵉⴳⴰ $f(x)=x^{2}$ , ⵜⴰⴼⵍⵡⵉⵜ ⵔⴰⴷ ⵜⴳ ⵉⵎⴽⴰ:

ⵜⴰⴼⵍⵡⵉⵜ ⵏ ⵡⴰⵣⴰⵍⵏ ⵉ: $f(x)=x^{2}$


$x$	$f(x)$
1	1
2	4
3	9
4	16

ⴰⵎⵣⵔⵓⵢ

ⵉⵏⵏⴼⵍⵉ ⵓⵔⵎⵎⵓⵙ ⵏ ⵜⵙⵖⵏⵜ ⴳ ⵓⵣⵎⵣ. ⵜⵓⴳⴳⵯⴰ ⴷ ⵜⵙⵡⵉⵏⴳⵎⵜ ⵜⴰⵎⵣⵡⴰⵔⵓⵜ ⵏ "ⵜⵓⵇⵇⵏⴰ ⵜⴰⵙⵖⵏⴰⵏⵜ" ⴳ ⵜⵡⵓⵔⵉⵡⵉⵏ ⵏ ⵉⵎⵙⵏⴰⴽⵜⵏ ⵉⴳⵔⵉⴽⵉⵢⵏ ⵉⵇⴱⵓⵔⵏ ⴳ ⵓⵙⴰⵜⴰⵍ ⵏ ⵜⵏⵣⴳⴳⵉⵜ, ⵎⴰⵛⴰ ⵓⵔ ⵜⴰ ⵜⵜ ⵉⵏⵏ ⵉⴽⴽⵉ ⵉⵍⵍⴰ ⵓⵙⵏⵎⵍ ⵏⵏⵙ ⴰⵎⴰⵜⴰⵢ. ⴳ ⵜⴰⵙⵓⵜ ⵜⵉⵙⵙ 14, ⵙⵙⵏⵜⵉⵏ ⵉⵎⵙⵡⵉⵏⴳⵎⵏ ⵣⵓⵏ ⴷ ⵏⵉⴽⵓⵍ ⵓⵔⵙⵎ ⴰⵙⵏⵓⵍⴼⵓ ⵏ ⵉⵎⵙⴽⴰⵏⵏ ⵏ ⵜⵓⵇⵇⵏⵉⵡⵉⵏ ⴳⵔ ⵜⵙⵎⴽⵜⵉⵡⵉⵏ ⵉⵜⵜⵏⴼⴰⵍⵏ, ⴰⵢⵍⵍⵉ ⵉⴳⴰⵏ ⵜⵉⵣⵡⵉⵔⵉ ⵜⴰⵎⵏⵣⵓⵜ ⵏ ⵜⴳⵏⵙⴻⵙⵜ ⵜⴰⵎⵙⴽⴰⵏⵜ ⵏ ⵜⵙⵖⵏⵉⵏ.^[8]

ⵜⴻⵜⵜⵓⵙⵎⵔⴰⵙ ⵜⴳⵓⵔⵉ "ⵜⴰⵙⵖⵏⵜ" ⵜⵉⴽⴽⵍⵜ ⵜⴰⵎⵣⵡⴰⵔⵓⵜ ⴷⴰⵔ ⴳⵓⵜⴼⵔⵉⴷ ⴼⵉⵍⵀⵍⵎ ⵍⵉⴱⵏⵜⵣ ⴳ 1673 ⴰⴼⴰⴷ ⴰⴷ ⵉⴳⵍⵎ ⵢⴰⵜ ⵜⵙⵎⴽⵜⴰ ⵉⵣⴷⵉⵏ ⴷ ⵢⴰⵏ ⵓⵎⴽⵏⴰⵡ, ⵣⵓⵏ ⴷ ⵜⵉⵖⵣⵉ ⵏⵏⵙ. ⴳ 1748, ⵉⴼⴽⴰ ⵍⵢⵓⵏⴰⵔⴷ ⵓⵍⵔ ⴰⵙⵏⵎⵍ ⴰⵎⵣⵡⴰⵔⵓ ⵢⴰⵥⵏ ⵉ ⵓⵙⵏⵎⵍ ⴰⵜⵔⴰⵔ: "ⵜⴰⵙⵖⵏⵜ ⵏ ⵜⵙⵎⴽⵜⴰ ⵜⴰⵎⵙⴽⵉⵍⵜ ⵜⴳⴰ ⵜⴰⵏⴼⴰⵍⵉⵜ ⵜⴰⵎⵚⵍⴰⴹⵜ ⵉⵜⵜⵓⵙⵏⴰⵢⵏ ⵙ ⴽⵔⴰ ⵏ ⵜⵖⴰⵔⴰⵙⵜ ⵙⴳ ⵜⵙⵎⴽⵜⴰ ⴰⵏⵏ ⵜⴰⵎⵙⴽⵉⵍⵜ ⴷ ⵉⵎⴹⴰⵏⵏ ⵏⵖ ⵜⵉⵙⵎⴽⵜⵉⵡⵉⵏ ⵉⵣⴳⴰⵏ."^[9] ⵉⵎⴰ ⴰⵙⵏⵎⵍ ⴰⵜⵔⴰⵔ, ⵉⴱⴷⴷⵏ ⵖⴼ ⵜⵥⵕⵉ ⵏ ⵜⴳⵔⵓⵎⵎⵉⵡⵉⵏ, ⵉⵙⵔⵙ ⵜ ⵢⵓⵀⴰⵏ ⴱⵉⵜⵔ ⴳⵓⵙⵜⴰⴼ ⵍⵓⵊⵓⵏ ⴷⵉⵔⵉⵛⵍⵉⵜ ⴳ 1837, ⵙⴼⴰⵍⴽⵉⵏ ⵜ ⵉⵎⵙⵏⴰⴽⵜⵏ ⵢⴰⴹⵏⵉⵏ ⴳ ⵜⴳⵉⵔⴰ ⵏ ⵜⴰⵙⵓⵜ ⵜⵉⵙⵙ 19.

ⵜⵉⴼⵔⵉⵙⵉⵏ ⵜⵉⵙⵉⵍⴰⵏⵉⵏ

ⵥⴹⴰⵕⵏⵜ ⵜⵙⵖⵏⵉⵏ ⴰⴷ ⵜⵜⵓⵙⵉⵙⵎⵍⵏⵜ ⴳ ⵓⵏⵛⵜ ⵏ ⵜⴼⵔⵉⵙⵉⵏ ⵏⵏⵙⵏⵜ. ⵜⵉⴼⵔⵉⵙⵉⵏ ⴰⴷ ⵜⵉⵣⴰⵍⴰⵏⵉⵏ ⴳⴰⵏⵜ ⵜⴰⵢⵏⴳⵉⵔⵜ, ⵜⴰⴼⵍⴳⵉⵔⵜ, ⴷ ⵜⵙⵏⴳⵉⵔⵜ.

ⵜⴰⵙⵖⵏⵜ ⵜⴰⵢⵏⴳⵉⵔⵜ: ⴷⴰ ⵜⴻⵜⵜⴳⴳⴰ ⵜⵙⵖⵏⵜ f ⵜⴰⵢⵏⴳⵉⵔⵜ (ⵏⵖ "ⵢⴰⵏ ⵉ ⵢⴰⵏ") ⵉⴳ ⵉⵣⴷⵉ ⴽⵓ ⴰⴼⵔⴷⵉⵙ ⴳ ⵜⵓⴳⵏⴰ (ⴰⵣⵉⵍⴰⵍ) ⴷ ⵢⴰⵏ ⵓⴼⵔⴷⵉⵙ ⴰⵎⵢⵉⵡⵏ ⴳ ⵜⴰⵖⵓⵍⵜ. ⵙ ⵜⵖⴰⵔⴰⵙⵜ ⵢⴰⴹⵏⵉⵏ, ⵉⴳ f(x₁) = f(x₂), ⵢⵓⵛⵛⵍ ⴰⴷ ⵉⴳ x₁ = x₂. ⴰⵎⴷⵢⴰ: f(x) = 2x ⵜⴳⴰ ⵜⴰⵢⵏⴳⵉⵔⵜ.
ⵜⴰⵙⵖⵏⵜ ⵜⴰⴼⵍⴳⵉⵔⵜ: ⵜⴳⴰ ⵜⵙⵖⵏⵜ f: X → Y ⵜⴰⴼⵍⴳⵉⵔⵜ (ⵏⵖ "ⵖⴼ") ⵉⴳ ⵜⴳⴰ ⵜⵓⴳⵏⴰ ⵏⵏⵙ ⵜⴰⵣⴷⴰⵖⵓⵍⵜ Y ⵙ ⵜⵉⵎⵎⴰⴷ. ⴰⵏⴰⵎⴽ ⵏⵏⵙ ⵎⴰⵙ ⴷ ⵉ ⴽⵓ ⴰⴼⵔⴷⵉⵙ y ⴳ Y, ⵉⵍⵍⴰ ⴰⵡⴷ ⵢⴰⵏ ⵓⴼⵔⴷⵉⵙ x ⴳ X ⵍⵍⵉ ⴳ ⵔⴰⴷ ⵢⵉⵍⵉ f(x) = y. ⴰⵎⴷⵢⴰ: f(x) = x³ ⵙⴳ ℝ ⵖⵔ ℝ ⵜⴳⴰ ⵜⴰⴼⵍⴳⵉⵔⵜ.
ⵜⴰⵙⵖⵏⵜ ⵜⴰⵙⵏⴳⵉⵔⵜ: ⵜⴳⴰ ⵜⵙⵖⵏⵜ ⵜⴰⵙⵏⴳⵉⵔⵜ ⵉⴳ ⵜⴳⴰ ⵜⴰⵢⵏⴳⵉⵔⵜ ⴷ ⵜⴼⵍⴳⵉⵔⵜ ⴳ ⵢⴰⵏ ⵡⴰⴽⵓⴷ. ⴰⵏⴰⵎⴽ ⵏⵏⵙ ⵎⴰⵙ ⴷ ⴽⵓ ⴰⴼⵔⴷⵉⵙ ⴳ ⵜⴰⵖⵓⵍⵜ ⵉⵣⴷⵉ ⴷ ⵢⴰⵏ ⵓⴼⵔⴷⵉⵙ ⴰⵎⵢⵉⵡⵏ ⴳ ⵜⵣⴷⴰⵖⵓⵍⵜ, ⴷ ⴽⵓ ⴰⴼⵔⴷⵉⵙ ⴳ ⵜⵣⴷⴰⵖⵓⵍⵜ ⵉⵣⴷⵉ ⴷ ⵢⴰⵏ ⵓⴼⵔⴷⵉⵙ ⴰⵎⵢⵉⵡⵏ ⴳ ⵜⴰⵖⵓⵍⵜ. ⵜⵉⵙⵖⵏⵉⵏ ⵜⵉⵙⵏⴳⵉⵔⵉⵏ ⵖⵓⵔⵙⵏⵜ ⵜⵉⵙⵖⵏⵉⵏ ⵜⵉⵎⵙⴰⵖⵓⵍⵉⵏ. ⴰⵎⴷⵢⴰ: f(x) = x + 1 ⵜⴳⴰ ⵜⴰⵙⵏⴳⵉⵔⵜ ⵙⴳ ℝ ⵖⵔ ℝ.^[10]

ⴰⵏⴰⵡⵏ ⵉⵜⵜⵓⵙⵙⴰⵏⵏ ⵏ ⵜⵙⵖⵏⵉⵏ

ⵜⵉⵙⵖⵏⵉⵏ ⵜⵉⴳⵜⴼⵓⵍⵉⵏ: ⵜⵜⵓⵚⴽⴰⵏⵜ ⵜⵙⵖⵏⵉⵏ ⴰⴷ ⵙⴳ ⵜⵎⵔⵏⵉⵡⵜ ⵏ ⵉⵏⴼⵙⴰⵔⵏ ⵓⵎⵎⵉⴷⵏ ⵓⵎⵏⵉⴳⵏ ⵏ ⵓⵎⵙⴽⵉⵍ. ⵜⴰⵍⵖⴰ ⵜⴰⵎⴰⵜⴰⵢⵜ ⵏⵏⵙⵏⵜ ⵜⴳⴰ $f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\dots +a_{1}x+a_{0}$ .
ⵜⵉⵙⵖⵏⵉⵏ ⵜⵉⵖⵥⵏⴰⵏⵉⵏ: ⵜⵉⴷ ⴳⴰⵏⵜ ⴰⵙⵙⴰⵖ ⵏ ⵙⵏⴰⵜ ⵜⵙⵖⵏⵉⵏ ⵜⵉⴳⵜⴼⵓⵍⵉⵏ, $f(x)=P(x)/Q(x)$ , ⵍⵍⵉ ⴳ $Q(x)\neq 0$ .
ⵜⵉⵙⵖⵏⵉⵏ ⵜⵉⴽⵕⴷⵉⵙⴽⵜⴰⵏⵉⵏ: ⵣⵓⵏ ⴷ $sin(x)$ , $cos(x)$ , ⴷ $tan(x)$ . ⵜⵉⵙⵖⵏⵉⵏ ⴰⴷ ⵜⵜⵓⵙⵎⵔⴰⵙⵏⵜ ⴱⴰⵀⵔⴰ ⴳ ⵓⴳⵍⴰⵎ ⵏ ⵜⵓⵎⴰⵏⵉⵏ ⵜⵉⵡⴰⵍⴰⵏⵉⵏ.
ⵜⵉⵙⵖⵏⵉⵏ ⵜⵉⵙⵏⵏⴼⵙⴰⵔⵉⵏ: ⵜⵉⴷ ⵍⵍⴰⵏⵜ ⴳ ⵜⵍⵖⴰ $f(x)=a^{x}$ , ⵍⵍⵉ ⴳ ⵜⴳⴰ ⵜⵙⵉⵍⴰ a ⵉⵎⵣⴳⵉ ⵓⵎⵏⵉⴳ. ⴰⵏⵏⵔⵏⵉ ⵏⵏⵙⵏⵜ ⵉⵎⵎⵔⵉⵏ ⵉⴳⴰ ⵢⴰⵏ ⵙⴳ ⵜⴼⵔⵉⵙⵉⵏ ⵏⵏⵙⵏⵜ ⵜⵉⵔⵙⵍⴰⵏⵉⵏ.
ⵜⵉⵙⵖⵏⵉⵏ ⵜⵉⵍⵓⴳⴰⵔⵉⵜⵎⵉⵏ: ⵜⵉⴷ ⴳⴰⵏⵜ ⵜⵉⵙⵖⵏⵉⵏ ⵜⵉⵎⵙⴰⵖⵓⵍⵉⵏ ⵏ ⵜⵙⵖⵏⵉⵏ ⵜⵉⵙⵏⵏⴼⵙⴰⵔⵉⵏ. ⴷⴰ ⵜⵜⵓⵙⵎⵔⴰⵙⵏⵜ ⴳ ⵓⴼⵙⴰⵢ ⵏ ⵜⴳⴷⵉⵡⵉⵏ ⵍⵍⵉ ⴳ ⵉⵍⵍⴰ ⵓⵎⵙⴽⵉⵍ ⴳ ⵓⵏⴼⵙⴰⵔ.^[11]

ⵉⵙⴰⵖⵓⵍⵏ

^ Spivak, Michael (2008). Calculus (4th ed.). Publish or Perish. p. 39. ISBN 978-0-914098-91-1.
^ Halmos, Paul R. (1970). Naive Set Theory. Springer-Verlag. p. 30. ISBN 978-0-387-90092-6.
^ Bartle, Robert G. (1976). The Elements of Real Analysis (2nd ed.). Wiley. p. 19. ISBN 978-0-471-05464-1.
^ Eves, Howard (1990). An Introduction to the History of Mathematics (6th ed.). Saunders. p. 476. ISBN 978-0-03-029558-4.
^ Lang, Serge (1993). Real and Functional Analysis (3rd ed.). Springer-Verlag. p. 3. ISBN 978-0-387-94001-4.
^ Apostol, Tom M. (1974). Mathematical Analysis (2nd ed.). Addison-Wesley. p. 35. ISBN 978-0-201-00288-1.
^ "Function Graph -- from Wolfram MathWorld". mathworld.wolfram.com.
^ Boyer, Carl B.; Merzbach, Uta C. (2011). A History of Mathematics (3rd ed.). Wiley. p. 243. ISBN 978-0-470-52548-7.
^ Kleiner, Israel (2007). A History of Abstract Algebra. Birkhäuser. p. 55. ISBN 978-0-8176-4684-4.
^ Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). Wiley. pp. 2–3. ISBN 978-0-471-43334-7.
^ Stewart, James (2012). Calculus: Early Transcendentals (7th ed.). Cengage Learning. pp. 10–15. ISBN 978-0-538-49790-9.

[1] Spivak, Michael (2008). Calculus (4th ed.). Publish or Perish. p. 39. ISBN 978-0-914098-91-1.

[2] Halmos, Paul R. (1970). Naive Set Theory. Springer-Verlag. p. 30. ISBN 978-0-387-90092-6.

[3] Bartle, Robert G. (1976). The Elements of Real Analysis (2nd ed.). Wiley. p. 19. ISBN 978-0-471-05464-1.

[4] Eves, Howard (1990). An Introduction to the History of Mathematics (6th ed.). Saunders. p. 476. ISBN 978-0-03-029558-4.

[5] Lang, Serge (1993). Real and Functional Analysis (3rd ed.). Springer-Verlag. p. 3. ISBN 978-0-387-94001-4.

[6] Apostol, Tom M. (1974). Mathematical Analysis (2nd ed.). Addison-Wesley. p. 35. ISBN 978-0-201-00288-1.

[7] "Function Graph -- from Wolfram MathWorld". mathworld.wolfram.com.

[8] Boyer, Carl B.; Merzbach, Uta C. (2011). A History of Mathematics (3rd ed.). Wiley. p. 243. ISBN 978-0-470-52548-7.

[9] Kleiner, Israel (2007). A History of Abstract Algebra. Birkhäuser. p. 55. ISBN 978-0-8176-4684-4.

[10] Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). Wiley. pp. 2–3. ISBN 978-0-471-43334-7.

[11] Stewart, James (2012). Calculus: Early Transcendentals (7th ed.). Cengage Learning. pp. 10–15. ISBN 978-0-538-49790-9.

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