Here are the essential concepts you must grasp in order to answer the question correctly.
One-to-One Functions
A one-to-one function is a type of function where each output is produced by exactly one input. This means that if f(a) = f(b), then a must equal b. One-to-one functions have unique inverses, which is crucial for finding the inverse function f^-1(x). Understanding this property ensures that we can correctly derive and verify the inverse.
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Inverse Functions
An inverse function reverses the effect of the original function. For a function f(x), its inverse f^-1(x) satisfies the equations f(f^-1(x)) = x and f^-1(f(x)) = x. To find the inverse, we typically swap the roles of x and y in the equation and solve for y. This concept is essential for part a of the question, where we need to derive f^-1(x).
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Verification of Inverses
Verifying that two functions are inverses involves showing that applying one function to the result of the other returns the original input. This is done through the equations f(f^-1(x)) = x and f^-1(f(x)) = x. This verification is crucial for part b of the question, as it confirms the correctness of the derived inverse function.
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