MATH10-Q2-Lesson1.pptx GRADE 10 3rd Quarter
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- Composite functions combine other functions by substituting one into another.
- Inverse functions reverse the input and output of a function if it exists.
- Common functions that can be graphed include linear, quadratic, trigonometric, cubic, exponential and logarithmic functions.
Higher Maths 121 Sets And Functions 1205778086374356 2
The document discusses key concepts in set theory and functions, including:
- Sets can contain numbers, elements, and be represented using curly brackets.
- Venn diagrams use overlapping circles to show logical connections between sets.
- A function has a domain (input) and range (output), where each input is mapped to a unique output.
- Composite functions combine other functions by substituting one into another.
- Inverse functions reverse the input and output of a function if it exists.
- Common functions that can be graphed include linear, quadratic, trigonometric, cubic, exponential and logarithmic functions.
Lecture 2 family of fcts
This document provides an overview of various mathematical models and functions, including:
1) Linear, polynomial, power, rational, algebraic, trigonometric, exponential, and logarithmic functions.
2) It describes the key properties of each type of function, such as their domains, ranges, and graph shapes.
3) It also discusses how functions can be transformed through stretching, shrinking, shifting, and reflection.
5.2 primitive recursive functions
The document discusses primitive recursive functions and predicates. It defines primitive recursive functions as those that can be constructed from initial functions using only composition and recursion. Some examples of primitive recursive functions given are addition, multiplication, factorial, power, predecessor, absolute value, and bounded quantification. Predicates like equality, less than, negation, conjunction, disjunction, division, and primality are also shown to be primitive recursive. The concept of bounded minimalization is introduced, where a function returns the least value t for which a primitive recursive predicate P(t,x1,...xn) is true.
2 evaluating functions
This document discusses evaluating functions by substituting values for variables. It provides examples of evaluating various functions, including price functions. Students should be able to evaluate functions and solve problems involving function evaluation. Functions can be even, odd, or neither based on whether f(-x) = f(x) or f(-x) = -f(x). Examples are provided to identify functions as even, odd, or neither. Exercises include evaluating functions for given values and determining function parity.
A numerical method to solve fractional Fredholm-Volterra integro-differential...
The Goolden ratio is famous for the predictability it provides both in the microscopic world as well as in the dynamics of macroscopic structures of the universe. The extension of the Fibonacci series to the Fibonacci polynomials gives us the opportunity to use this powerful tool in the study of Fredholm-Volterra integro-differential equations. In this paper, we define a new hybrid fractional function consisting of block-pulse functions and Fibonacci polynomials (FHBPF). For this, in the Fibonacci polynomials we perform the transformation $x\to x^{\alpha}$, with $\alpha$ a real parameter. In the method developed in this paper, we propose that the unknown function $D^{\alpha}f(x)$ be written as a linear combination of FHBPF. We consider the fractional derivative $D^{\alpha}$ in the Caputo sense. Using theoretical considerations, we can write both the function $f(x)$ and other involved functions of type $D^{\beta}f(x)$ on the same basis. For this operation, we have to define an integral operator of Riemann-Liouville type associated to FHBPF, and with the help of hypergeometric functions, we can express this operator exactly. All these ingredients together with the collocation in the Newton-Cotes nodes transform the integro-differential equation into an algebraic system that we solve by applying Newton's iterative method. We conclude the paper with some examples to demonstrate that the proposed method is simple to implement and accurate. There are situations when by simply considering $\alpha\ne1$, we obtain an improvement in accuracy by 12 orders of magnitude.
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- 2. Definition of Polynomial Function •
- 4. Example: P(x) = 4x³ - 2x² + x - 5 Leading coefficient Degree Leading term Constant term
- 6. EXAMPLES OF POLYNOMIAL FUNCTION •
- 7. NOT A POLYNOMIAL FUNCTION •
- 8. KINDS OF POLYNOMIAL FUNCTIONS Polynomial Function Examples Degree Leading Coefficient Constant Term Zero Function f(x) = 0 None None None Constant Function f(x) = 2 0 0 2 Linear Function g(x) = x - 7 1 1 -7 Quadratic Function h(x) = 5x² + 9 2 5 9 Cubic Function p(x) = x³ - 2x² - 4x + 6 3 1 6 Quartic Function Q(x) = 3x⁴ + 5x³ - x - 8 4 3 -8 Quintic Function R(x) = 2x⁵ - x⁴ - 4x 5 2 0 nth Degree Polynomial Function F(x) = 7x⁶ + 5x⁴ - 3 6 7 -3
- 9. Polynomial Polynomial Function Degree Leading Coefficien t Constan t Term 1. p(x) = 1 2 x – 4x² + 2 2. h(x) = x(2x – 3)² 3. g(x) = -2x⁴ + 7x + 1 4. y = 7.5x¹⁰ - 3x⁴ + 11x⁸ 5. H(x) = 1 2 x² + 4x - 5 Complete the table below.