A) Does the function f(x)=|x| have an inverse? Explain. b)...

a) Does the function $f(x)=|x|$ have an inverse? Explain.
b) If the domain is limited to $x \geq 0$, does the function have an inverse? Explain.
c) Determine the inverse function of $f(x)=|x|, x \geq 0$.

Key Concepts

Domain Restriction

Domain restriction involves limiting the set of input values for which a function is considered. When a function is not one-to-one over its entire domain, imposing a domain restriction can help make the function invertible. By restricting the domain to a subset where the function is one-to-one, it becomes possible to define a unique inverse function.

Inverse Function

An inverse function reverses the operation of the original function, essentially swapping the roles of inputs and outputs. For a function to have an inverse, it must be one-to-one on its domain. If a function is not one-to-one, an inverse cannot be uniquely defined over its entire domain unless additional restrictions are applied.

Absolute Value Function

The absolute value function, often written as |x|, outputs the non-negative magnitude of a real number. This means regardless of whether x is positive or negative, the function returns a non-negative value. It is symmetric about the y-axis because for every positive x, the function value is the same as for -x, which is a key consideration when determining its invertibility.

One-to-One Function

A function is one-to-one (injective) if every output is produced by exactly one input. This property is critical for a function to have an inverse because an inverse function must reverse the action of the original function uniquely. If a function maps two or more different inputs to the same output, it violates the condition needed for strict invertibility.

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