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

Key Concepts

Absolute Value Function

The absolute value function, denoted |x|, returns the non-negative magnitude of a real number x. It is characterized by its 'V'-shaped graph, which is symmetric about the y-axis, meaning it produces the same output for x and -x. This inherent symmetry is why the function is not one-to-one over the set of all real numbers, as multiple inputs yield the same output.

Horizontal Line Test

The horizontal line test is a graphical method used to determine if a function is one-to-one. If every horizontal line intersects the graph of the function at most once, then the function is one-to-one. This test provides a visual criterion for assessing injectivity and is a useful tool when analyzing whether an inverse function exists.

One-to-One Functions

A function is one-to-one (or injective) if every element of its range is mapped by exactly one element of its domain. This means that if f(a) = f(b), then a must equal b. An understanding of one-to-one functions is crucial because only one-to-one functions have well-defined inverses that can 'undo' the operation of the original function.

Inverse Functions

An inverse function essentially reverses the action of the original function. If f is a one-to-one function, its inverse, denoted by f?¹, satisfies the conditions f(f?¹(x)) = x and f?¹(f(x)) = x for every x in the appropriate domain and range. Understanding how to determine and compute inverse functions is key in many areas of mathematics, particularly when solving equations or modeling reversible processes.

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