Transcription of Lecture 1 - UH
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Lecture 1 - UH
Lecture 1 Section One-To-One Functions; InversesJiwen He1 One-To-One Definition of the One-To-One FunctionsWhat are One-To-One Functions? Geometric TestHorizontal Line Test If some horizontal line intersects the graph of the function more than once,then the function is not one-to-one. If no horizontal line intersects the graph of the function more than once,then the function is are One-To-One Functions? Algebraic TestDefinition functionfis said to beone-to-one(or injective) iff(x1) =f(x2) impliesx1= functionfis one-to-one if and only if x1, x2,x16=x2impliesf(x1)6=f(x2).1 Examples and Counter-ExamplesExamples3. f(x) = 3x 5 is 1-to-1. f(x) =x2is not 1-to-1. f(x) =x3is 1-to-1. f(x) =1xis 1-to-1.
Examples and Counter-Examples Examples 3. • f(x) = 3x−5 is 1-to-1. • f(x) = x2 is not 1-to-1. • f(x) = x3 is 1-to-1. • f(x) = 1 x is 1-to-1. • f(x) = xn −x, n > 0, is not 1-to-1. Proof. • f(x 1) = f(x 2) ⇒ 3x 1 − 5 = 3x 2 − 5 ⇒ x 1 = x 2.In general, f(x) = ax−b, a 6= 0, is 1-to-1.
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