ss_4_1

ss_4_1 - Section 4.1 One-to-One Functions A function f is...

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Section 4.1 One-to-One Functions A function, f ,is one-to-one if f ( x 1 )= f ( x 2 ) implies x 1 = x 2 , i.e., if x 1 6 = x 2 ,then f ( x 1 ) 6 = f ( x 2 ). Example. The function f ( x )=2 x +1 is one-to-one because f ( x 1 f ( x 2 ) implies 2 x 1 +1 = 2 x 2 +1 and this implies that x 1 = x 2 . The graph of f is shown below. Note that all horizontal lines intersects the graph of f at one point. –4 –2 2 4 –4 –2 2 4 f ( x x Example. f ( x x 2 is not one-to-one. For example, if x 1 =1and x 2 = - 1, then x 1 6 = x 2 but f ( x 1 f ( x 2 ). The graph of f is shown below. Note that some horizontal lines intersect the graph of f at more than one point. –4 –2 2 4 –4 –2 2 4 f ( x x 2 The above two graphs illustrate the following horizontal line test. Horizontal Line Test. A function f is one-to-one iﬀ every horizontal line intersects the graph of f in at most one point. A function f is increasing if f ( a ) <f ( b ) whenever a<b and a and b are in the domain of f .F o r example, the function given by f ( x )=3 x + 2 is an increasing function. A function f is decreasing if f ( a ) >f ( b ) whenever and a and b are in the domain of f . For example, the function given by f ( x - 2 x + 3 is a decreasing function.

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ss_4_1 - Section 4.1 One-to-One Functions A function f is...

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