### Function Analysis: Well-Defined CriteriaLet's determine whether the given functions are well-defined, considering their domains and any restrictions imposed by their expressions.#### (a) Function $ f: \mathbb{R} \to \mathbb{R} $- **Definition:** $ f(x) = \frac{3}{x^2 + 5} $- **Domain:** The function is defined for all real numbers $ x $ because the denominator $ x^2 + 5 $ is always positive and never zero (since $ x^2 \geq 0 $ implies $ x^2 + 5 \geq 5 $).- **Conclusion:** The function $ f(x) $ is well-defined over the entire set of real numbers.#### (b) Function $ g: (5, \infty) \to \mathbb{R} $- **Definition:** $ g(x) = \frac{7}{\sqrt{x - 4}} $- **Domain:** The expression $ \sqrt{x - 4} $ implies $ x - 4 > 0 $, so $ x > 4 $. Thus, the domain is properly set as $ (5, \infty) $ because each $ x $ in this interval satisfies $ x > 4 $.- **Conclusion:** The function $ g(x) $ is well-defined on $ (5, \infty) $.#### (c) Function $ h: \mathbb{R} \to \mathbb{R} $- **Definition:** $ h(x) = -\sqrt{7 - 4x + 4x^2} $- **Domain:** The expression inside the square root $ 7 - 4x + 4x^2 $ must be non-negative for $ h(x) $ to be defined. Calculating the discriminant of the quadratic $ 4x^2 - 4x + 7 $ shows that it doesn't have real roots, and therefore, it is always positive.- **Conclusion:** The function $ h(x) $ is well-defined for all real numbers.Each function meets the criteria for being well-defined within its specified domain.

Answered: Image /qna-images/answer/b8aab230-abda-4b13-9c9f-264bd637cab1.jpg

Answered: 2. Determine whether these are…

2. Determine whether these are well-defined functions. Explain. 3 (a) f:R → R, where f(x) = 2² + 5° 7 (b) g: (5, 0) → R, where g(x) = Vx – 4 (c) h:R → R, where h(x) = -V7 – 4x + 4x².

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.

Question

### Function Analysis: Well-Defined Criteria

Let's determine whether the given functions are well-defined, considering their domains and any restrictions imposed by their expressions.

#### (a) Function $ f: \mathbb{R} \to \mathbb{R} $

- **Definition:** $ f(x) = \frac{3}{x^2 + 5} $
- **Domain:** The function is defined for all real numbers $ x $ because the denominator $ x^2 + 5 $ is always positive and never zero (since $ x^2 \geq 0 $ implies $ x^2 + 5 \geq 5 $).
- **Conclusion:** The function $ f(x) $ is well-defined over the entire set of real numbers.

#### (b) Function $ g: (5, \infty) \to \mathbb{R} $

- **Definition:** $ g(x) = \frac{7}{\sqrt{x - 4}} $
- **Domain:** The expression $ \sqrt{x - 4} $ implies $ x - 4 > 0 $, so $ x > 4 $. Thus, the domain is properly set as $ (5, \infty) $ because each $ x $ in this interval satisfies $ x > 4 $.
- **Conclusion:** The function $ g(x) $ is well-defined on $ (5, \infty) $.

#### (c) Function $ h: \mathbb{R} \to \mathbb{R} $

- **Definition:** $ h(x) = -\sqrt{7 - 4x + 4x^2} $
- **Domain:** The expression inside the square root $ 7 - 4x + 4x^2 $ must be non-negative for $ h(x) $ to be defined. Calculating the discriminant of the quadratic $ 4x^2 - 4x + 7 $ shows that it doesn't have real roots, and therefore, it is always positive.
- **Conclusion:** The function $ h(x) $ is well-defined for all real numbers.

Each function meets the criteria for being well-defined within its specified domain.

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