For each function, a) determine whether it is one-to-one; b)...

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Key Concepts

Inverse Functions

The inverse function reverses the roles of inputs and outputs, meaning that if f maps an input x to an output y, then the inverse function, denoted as f?¹, maps y back to x. Finding the inverse typically involves interchanging the roles of x and y in the equation and then solving for y. For functions that are one-to-one, like increasing cubic functions, the inverse exists and it 'undoes' the effect of the original function.

One-to-One Functions

A one-to-one function (or injective function) is a function in which every output is produced by exactly one input, meaning that if f(a) equals f(b), then a must equal b. This property ensures that the function does not map two different inputs to the same output, making it invertible. For functions like a cubic function that is monotonic (strictly increasing or decreasing), proving one-to-one-ness usually relies on this monotonicity argument, as no horizontal line intersects the graph more than once.

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