Скачать или смотреть Making x^2 One-to-One: The Power of DOMAIN RESTRICTIONS

TL;DW: A standard parabola like f(x) = x² + 4 fails the Horizontal Line Test because it is NOT one-to-one. To find an inverse function f⁻¹, we must restrict the domain to, most commonly, D = [0, ∞). This keeps only the right "half" of the parabola, making it invertible. Another common choice for a restricted domain is D = (-∞,0].

THE PROBLEM WITH QUADRATICS:

General Case: For example, f(1) = 5 and f(-1) = 5. Two different inputs give the same output. Therefore, f is NOT a one-to-one function over the entire real number line ℝ = (-∞,∞).

Failure: In other words, this fails the Horizontal Line Test (HLT).

Result: No inverse exists for the full parabola.

THE SOLUTION (DOMAIN RESTRICTION):

By setting the domain to [0, ∞), we ignore negative x-values.

The remaining graph is strictly increasing.

It now passes the Horizontal Line Test and is ONE-TO-ONE.

COMING UP NEXT: Part 6 - Solving for the inverse of f(x) = x² + 4. Why the restricted domain forces us to choose the POSITIVE square root!

#InverseFunctions #DomainRestriction #OneToOneFunction #Algebra2 #Precalculus #MathTutorial #Parabola #HorizontalLineTest

RESTRICTING the DOMAIN can produce a one-to-one function from one that is not one-to-one

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