If f is a one-to-one function and f(4) = 5, f(6) = 7, and f(13) = −10, find f −1(7), f −1(5), and f −1(−10).SolutionFrom the definition of f −1 we havef −1(7) = because f = 7f −1(5) = because f = 5f −1(−10) = because f = −10The diagrams make it clear how f −1 reverses the effect of f in this case.4613There are two sets, the set A contains 3 elements, the set B contains 3 elements, and 3 arrows connect the set A to the set B, the relation from set A to set B is labeled f.The set A contains the elements 4, 6 and 13.The set B contains the elements 5, 7 and −10.An arrow connects 4 in the set A to 5 in the set B.An arrow connects 6 in the set A to 7 in the set B.An arrow connects 13 in the set A to −10 in the set B.4613There are two sets, the set B contains 3 elements, the set A contains 3 elements, and 3 arrows connect the set B to the set A, the relation from set B to set A is labeled f −1.The set B contains the elements 5, 7 and −10.The set A contains the elements 4, 6 and 13.An arrow connects 5 in the set B to 4 in the set A.An arrow connects 7 in the set B to 6 in the set A.An arrow connects −10 in the set B to 13 in the set A.
Question
If f is a one-to-one function and f(4) = 5, f(6) = 7, and f(13) = −10, find f −1(7), f −1(5), and f −1(−10).SolutionFrom the definition of f −1 we havef −1(7) = because f = 7f −1(5) = because f = 5f −1(−10) = because f = −10The diagrams make it clear how f −1 reverses the effect of f in this case.4613There are two sets, the set A contains 3 elements, the set B contains 3 elements, and 3 arrows connect the set A to the set B, the relation from set A to set B is labeled f.The set A contains the elements 4, 6 and 13.The set B contains the elements 5, 7 and −10.An arrow connects 4 in the set A to 5 in the set B.An arrow connects 6 in the set A to 7 in the set B.An arrow connects 13 in the set A to −10 in the set B.4613There are two sets, the set B contains 3 elements, the set A contains 3 elements, and 3 arrows connect the set B to the set A, the relation from set B to set A is labeled f −1.The set B contains the elements 5, 7 and −10.The set A contains the elements 4, 6 and 13.An arrow connects 5 in the set B to 4 in the set A.An arrow connects 7 in the set B to 6 in the set A.An arrow connects −10 in the set B to 13 in the set A.
Solution
The inverse function, denoted as f −1, essentially reverses the operation of the original function. Given that f is a one-to-one function, each input corresponds to a unique output and vice versa.
From the given function, we know that: f(4) = 5 f(6) = 7 f(13) = -10
To find the inverse, we simply reverse the inputs and outputs.
So, f −1(7) = 6 because f(6) = 7 f −1(5) = 4 because f(4) = 5 f −1(-10) = 13 because f(13) = -10
The diagrams mentioned in the question would visually represent these relationships, showing how f −1 reverses the effect of f.
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