Table of Contents
Introduction to Real Numbers
Real numbers are those that can be expressed in decimal form, and include all rational and irrational numbers. Rational numbers are those that can be expressed as a fraction, while irrational numbers cannot be expressed as a fraction. All real numbers can be expressed in decimal form, and include both rational and irrational numbers.
What is a Real Function?
A real function is a mathematical function that assigns a real number to every point in a given set. In other words, it is a set of ordered pairs (x, y) where each x corresponds to a unique y. A real function can be represented using a graph, which will show the relationship between the x and y values.
Properties of Positive Real Function
A positive real function is a function that has a real domain and a real range, and all of the values in the range are positive. Additionally, a positive real function is always increasing, meaning that as you move from one point on the function to another point on the function, the value of the function always increases.
Operations on Real Functions Inverse Functions
The inverse of a function is a function that “undoes” the original function. Given a function f, the inverse of f is denoted f-1. The inverse of a function is usually not the same as the function it is inverse to.
There are several properties of inverse functions that must be satisfied in order for a function to have an inverse. First, the function must be one-to-one. This means that for every input there is only one output. Second, the function must be onto. This means that the range of the function must be the same as the domain.
In order to find the inverse of a function, we first need to graph the function. Once the function is graphed, we can find the inverse by using the inverse function theorem. The inverse function theorem states that if the function is one-to-one and onto, then the inverse function exists and is unique.
To find the inverse of a function, we must first determine its function notation. We do this by taking the function and flipping it upside down. So, if f(x) = 5x + 3, then f-1(x) = 3x + 5.
Next, we must determine the domain and range of the inverse function. The domain of the inverse function is the range of the original function. The range of the inverse function is the domain of the original function.
Finally,
Solved Example
Q. A metal box of dimensions 10 cm × 15 cm × 20 cm is to be made. What is the length of metal required?
Solution:
The length of metal required is the sum of the lengths of the three sides of the box.
10 cm + 15 cm + 20 cm = 45 cm
What is a Real Function?
A function whose range lies within the real numbers i.e., non-root numbers and non-complex numbers, is said to be a real function, also called a real-valued function.
A real function is an entity that assigns values to arguments. The notation P = f (x) means that to the value x of the argument, the function f assigns the value P. Sometimes, we also use the notation f: x ↦ P, in words, the function f sends x to P. The most usual way of specifying this assignment is by some formula, that is, the function value P can be obtained by substituting x to a specific formula that identifies the given function.
Any function in the form of F(x) is called a positive real function, if it falls under these four critical categories:
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F(v) should have real values for all real values of x.
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F(v) must be a Hurwitz polynomial.
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If we substitute v = j*ω then on splitting up the real and imaginary parts, the real part of the function must be more than or equivalent to zero which means it should not be negative. This is the most critical condition, and we frequently use this theory to clear doubts regarding the fact that the function is a positive real function or not.
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On substituting v = go, F(x) should own simple poles, and the residues must be real and positive.
