Use Definition 0.2 to prove that the range of the function...

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Key Concepts

Proof Techniques in Real Analysis

Proof techniques in real analysis, such as direct proofs, are used to establish properties of functions rigorously. They involve starting with the definitions and logically deducing the desired conclusions. This methodical approach is essential when verifying that a function’s range covers an entire set, such as the set of all real numbers.

Range of a Function

The range of a function is the set of all output values that the function can produce when applied to every element in its domain. It provides a complete description of the outputs and is a fundamental concept in understanding how functions map inputs to outputs.

Surjective (Onto) Functions

A function is called surjective, or onto, if every element in the codomain is the image of at least one element from the domain. Surjectivity is a key property when proving that the range of a function equals a particular set, as it ensures that no element in the target set is left unmapped.

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