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One-to-One Functions; Inverse Function

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A function f is one-to-one if for each x in the domain of f there is exactly one y in the range and no y in the range is the image of more than one x in the domain. A function is not one-to-one if two different elements in the domain correspond to the same element in the range.

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x1x1 x2x2 x3x3 y1y1 y2y2 y3y3 x1x1 x3x3 y1y1 y2y2 y3y3 x1x1 x2x2 x3x3 y1y1 y3y3 Domain Range One-to-one function Not a function NOT One-to-one function

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M : Mother Function is NOT one-one Joe Samantha Anna Ian Chelsea George Laura Julie Hilary Barbara Sue HumansMothers

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S: Social Security function IS one-one Joe Samantha Anna Ian Chelsea George 123456789 223456789 333456789 433456789 533456789 633456789 AmericansSSN

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Is the function f below one – one? 12345671234567 10 11 12 13 14 15 16

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Theorem Horizontal Line Test If horizontal lines intersect the graph of a function f in at most one point, then f is one-to-one.

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Use the graph to determine whether the function is one-to-one. Not one-to-one.

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Use the graph to determine whether the function is one-to-one. One-to-one.

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The inverse of a one-one function is obtained by switching the role of x and y

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123456789 223456789 333456789 433456789 533456789 633456789 SSN Joe Samantha Anna Ian Chelsea George Americans The inverse of the social security function

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Let and Find

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g is the inverse of f.

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. Let f denote a one-to-one function y = f(x). The inverse of f, denoted by f -1, is a function such that f -1 (f( x )) = x for every x in the domain of f and f(f -1 (x))=x for every x in the domain of f -1.

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Domain of fRange of f

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Theorem The graph of a function f and the graph of its inverse are symmetric with respect to the line y = x.

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y = x (2, 0) (0, 2)

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Finding the inverse of a 1-1 function Step1: Write the equation in the form Step2: Interchange x and y. Step 3: Solve for y. Step 4: Write for y.

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Find the inverse of Step1: Step2: Interchange x and y Step 3: Solve for y

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