# **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.
* **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 $f^{-1}(f(x)) = x$. --- # **Important Formulas** Below are the key formulas and conditions used in this exercise: | Concept | Mathematical Notation / Condition | | :--- | :--- | | **Domain of Rational Function** | For $f(x) = \frac{P(x)}{Q(x)}$, $Q(x) \neq 0$ | | **Domain of Radical Function** | For $f(x) = \sqrt{g(x)}$, $g(x) \geq 0$ | | **One-to-One Condition** | If $f(x_1) = f(x_2) \implies x_1 = x_2$ | | **Bijective Condition** | Function is both Injective and Surjective |
| **Inverse Verification** | $f(f^{-1}(x)) = f^{-1}(f(x)) = x$ | | **Sine Function Range** | $-1 \leq \sin(x) \leq 1$ |
--- # **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 $|x| \geq 0$ or $x^2 \geq 0$. 3. **For Inverses:** Systematic replacement of $f(x)$ with $y$, swapping variables, and verifying the result ensures accuracy.