M1410109 - (Section 1.9: Inverse Functions) 1.99 SECTION...

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(Section 1.9: Inverse Functions) 1.99 SECTION 1.9: INVERSE FUNCTIONS PART A: ONE-TO-ONE (“1-1”) FUNCTIONS; HORIZONTAL LINE TEST (HLT) A function is one-to-one (“1-1”) No two inputs from its domain yield the same output. Remember that a function by definition cannot have an input from its domain that yields two or more different outputs. See Section 1.4, Notes 1.19 . The graph of y = f x ( ) represents a one-to-one (“1-1”) function f It passes both the Vertical Line Test (VLT) and the Horizontal Line Test (HLT) . If both tests are passed, then there is a one-to-one (“1-1”) correspondence between the inputs in the domain and the outputs in the range. Think of the x , y ( ) pairs as married couples that go together with no sharing. For example, you could have: ( x as temperature in degrees Fahrenheit, y as the same temperature in degrees Celsius). See Section 1.6, Notes 1.70 . Example If f x ( ) = x 2 , then f is not a one-to-one (“1-1”) function. It is assumed that the domain is R . Observe that the graph of f fails the HLT.
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(Section 1.9: Inverse Functions) 1.100 It is easy to find two x inputs in the domain that yield the same y output. For example, However, if we restrict the domain to 0, ) , the nonnegative reals, then f is a one-to-one function. Note: We will use the idea of restricted domains when we define inverse trig functions in Section 4.7 . Observe that the graph of f now passes the HLT. There is now a one-to-one (“1-1”) correspondence between the x values in the domain and the y values in the range. (Think: Married couples. For example, x = 3 is married to y = 9 .)
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(Section 1.9: Inverse Functions) 1.101 PART B: THE INVERSE OF A FUNCTION A function has an inverse function (i.e., it is invertible) It is one-to-one (“1-1”). If
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M1410109 - (Section 1.9: Inverse Functions) 1.99 SECTION...

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