Exercise Questions

All questions in this exercise are listed below. Click on a question to view its solution.

Key Concepts

This exercise focuses on the following concepts:

• Domain and Range:

  • Domain: The set of all possible input values ($x$) for which the function is defined.

  • Range: The set of all possible output values ($y$) the function can produce.

• Types of Functions:

  • One-to-One (Injective): Every element in the codomain is mapped to by at most one element in the domain.

  • Onto (Surjective): The range is equal to the codomain.

  • Into: The range is a proper subset of the codomain.

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• Bijective: A function that is both One-to-One and Onto.

• Inverse Functions ($f^{-1}$):

  • A function that reverses the operation of $f$.

  • Relationship: $\text{Domain of }f=\text{Range of }{f}^{-1}$ and $\text{Range of }f=\text{Domain of }{f}^{-1}$.

  • Verification: $f(f^{-1}(x))=x$ and $f^{-1}(f(x))=x$.

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Important Formulas

Below are the key formulas and conditions used in this exercise:

| Domain of Radical Function | For $f(x)=\sqrt{g(x)}$, $g(x)\ge 0$ |

| Inverse Verification | $f(f^{-1}(x))=f^{-1}(f(x))=x$ |

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Summary

This exercise provides a comprehensive review of functional analysis. Key learnings include identifying points of discontinuity in rational functions to determine Domain and utilizing algebraic manipulation to solve for $x$ in terms of $y$ to find the Range and Inverse.

Strategies for Success:

1. For Domain: Always look for denominators (cannot be zero) and square roots (radicand must be non-negative).
2. For Range: Consider the behavior of the function as $x\to \infty$ or look for constraints like $|x|\ge 0$ or $x^2\ge 0$.
3. For Inverses: Systematic replacement of $f(x)$ with $y$, swapping variables, and verifying the result ensures accuracy.