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. If f(x) takes an input x and produces an output y, then the inverse function f^(-1)(y) will take y back to x. To find the inverse, we typically solve the equation y = f(x) for x in terms of 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 Inverse Functions
To verify that two functions are inverses, we must show that applying one function to the result of the other returns the original input. Specifically, we check that f(f^(-1)(x)) = x and f^(-1)(f(x)) = x. This verification confirms the correctness of the derived inverse function and is a critical step in the problem-solving process.
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