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

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Key Concepts

Inverse Functions

An inverse function essentially reverses the effect of the original function by swapping the roles of the input and output. For a function to have an inverse, it must be one-to-one over its domain. The process of finding an inverse involves expressing the original function in the form y = f(x), then interchanging x and y and solving for y.

Quadratic Functions and Domain Restrictions

Quadratic functions, such as one defined by a squared term plus a constant, are typically not one-to-one when defined over the entire set of real numbers due to their symmetry about the vertex. However, by restricting the domain (for example, to non-negative numbers if the parabola opens upward), a quadratic function can become one-to-one, allowing the determination of an inverse function on that restricted domain.

One-to-One Functions

A function is one-to-one (injective) if every element in its domain maps to a unique element in its range, meaning no two distinct inputs have the same output. This property can be verified by checking that for any two values a and b, if the outputs f(a) and f(b) are equal, then a must equal b. Graphically, a function is one-to-one if every horizontal line crosses the graph at most once.

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