Piecewise function lesson 3
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This document discusses piecewise functions, which are functions defined by separate formulas on different pieces of the domain. It provides examples of piecewise functions and how to evaluate them by identifying which piece of the domain the input value falls in and using the corresponding formula. The key aspects are: - Piecewise functions use different rules for different intervals of the domain. - To evaluate, identify which interval contains the input and use that rule. - Examples show how to write the piecewise function rule and evaluate it for given values.
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This document discusses linear programming and optimization. It begins with essential questions about finding maximum and minimum values of functions over regions. Key vocabulary is defined, including linear programming, feasible region, bounded, unbounded, and optimize. Two examples are provided to demonstrate how to graph inequality systems, identify feasible regions, and find the maximum and minimum values of an objective function over those regions using linear programming techniques.
Functions
The document discusses functions and relations. It defines functions, relations, and domain and range. It provides examples of expressing relations in set notation, tabular form, equations, graphs, and mappings. It also discusses evaluating, adding, multiplying, dividing, and composing functions. Graphs of various functions like absolute value, piecewise, greatest integer, and least integer functions are also explained.
Derivatives
This document provides information about derivatives and their applications:
1. It defines the derivative as the limit of the difference quotient, and explains how to calculate derivatives using first principles. It also covers rules for finding derivatives of sums, products, quotients, exponentials, and logarithmic functions.
2. Higher order derivatives are introduced, with examples of how to take second and third derivatives.
3. Applications of derivatives like finding velocity and acceleration from a position-time function are demonstrated. Maximum/minimum values and how to find local and absolute extrema are also discussed with an example.
2.6 Graphs of Basic Functions
This document provides information about graphing basic functions such as the identity, squaring, cubing, square root, cube root, and absolute value functions. It discusses concepts of continuity and discontinuity. It also describes how to graph piecewise functions, including the greatest integer function, using Desmos. Key functions and their domains, ranges, and graphs are defined.
7 functions
This document defines functions and relations. It discusses identifying the domain and range of functions and relations, evaluating functions, and performing operations on functions such as addition, subtraction, multiplication, division, and composition. It also covers graphing functions, including piecewise functions, absolute value functions, greatest and least integer functions. Key examples are provided to illustrate how to identify domains and ranges, evaluate functions, perform operations on functions, and graph different types of functions.
Lesson 2 - Functions and their Graphs - NOTES.ppt
The document provides lesson material on functions and their graphs. It includes sections on defining functions, determining if a relation is a function, functional notation, domain and range, graphing functions, and identifying intervals of increase/decrease. Additional topics covered are relative min/max values, step functions, even and odd functions, and piecewise-defined functions. Examples and exercises are provided to illustrate key concepts.
3. Functions II.pdf
Properties of Functions
Odd and Even Functions
Periodic Functions
Monotonic Functions
Bounded Functions
Maxima and Minima of Functions
Inverse Function
Sequence and Series
1 5 graphs of functions
The document discusses graphs of functions and their properties. It defines a function graph as a collection of ordered pairs (x, f(x)) where x is in the function's domain. A function's domain is the set of x-values where it is defined, and its range is the set of corresponding y-values. A relation is a function if its graph passes the vertical line test, meaning no vertical line intersects the graph at more than one point. Even and odd functions are defined as those where f(-x) equals f(x) or -f(x), respectively.
Note introductions of functions
- A function is a rule that maps an input number (independent variable) to a unique output number (dependent variable).
- To determine if a rule describes a valid function, you can plot points from the rule on a graph and check that each input only maps to one output using a vertical ruler.
- For a rule to describe a valid function, its domain must be restricted if multiple outputs are possible for any single input. The domain is the set of possible inputs, and the range is the set of corresponding outputs.
Introduction to functions
- A function is a rule that maps an input number (independent variable) to a unique output number (dependent variable).
- To determine if a rule describes a valid function, you can plot points from the rule on a graph and check that each input only maps to one output using a vertical ruler.
- For a rule to describe a valid function, its domain must be restricted if multiple outputs are possible for any single input. The domain is the set of possible inputs, and the range is the set of corresponding outputs.
Mathematics 9 Quadratic Functions (Module 1)
This learner's module discusses or talks about the topic of Quadratic Functions. It also discusses what is Quadratic Functions. It also shows how to transform or rewrite the equation f(x)=ax2 + bx + c to f(x)= a(x-h)2 + k. It will also show the different characteristics of Quadratic Functions.
Module 1 quadratic functions
This module introduces quadratic functions. It discusses identifying quadratic functions as those with the highest exponent of 2, rewriting quadratic functions in general form f(x) = ax^2 + bx + c and standard form f(x) = a(x-h)^2 + k, and the key properties of quadratic graphs including the vertex and axis of symmetry. The module provides examples of identifying quadratic functions from equations and ordered pairs/tables and rewriting quadratic functions between general and standard form using completing the square.
1531 fourier series- integrals and trans
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Introduction to Functions
- A function is a rule that maps each input to a unique output. Not every rule defines a valid function.
- For a rule to be a valid function, it must map each input to only one output. The domain is the set of valid inputs, and the range is the set of corresponding outputs.
- Functions can be represented graphically by plotting the input-output pairs. The graph of a valid function should only intersect the vertical line above each input once.
5.1 Quadratic Functions
* Recognize characteristics of parabolas.
* Understand how the graph of a parabola is related to its quadratic function.
* Determine a quadratic function’s minimum or maximum value.
* Solve problems involving a quadratic function’s minimum or maximum value.
Lesson 4A - Inverses of Functions.ppt
This document discusses inverses of functions. It begins by introducing the concept of inverses and using the story of the Sneetches to provide an example. It then discusses how to find the inverse of a function by switching the x's and y's and solving for y. Several examples are worked through. It emphasizes that a function must be one-to-one to have an inverse and discusses using the horizontal line test to determine if a function is one-to-one. The document concludes by discussing how the composition of a function and its inverse will result in the original input and exploring examples of functions that are inverses of each other.
Calculus - 1 Functions, domain and range
This document defines functions and discusses key concepts related to functions including:
- A function relates each element of its domain to a unique element of its range.
- Functions can be one-to-one or many-to-one.
- Functions are represented in set notation, tabular form, equations, and graphs.
- The domain of a function is the set of possible inputs, and the range is the set of possible outputs.
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Piecewise function lesson 3
- 2. Piecewise Functions Objectives: Become familiar with piecewise functions Evaluate piecewise functions
- 4. Piecewise Functions A piecewise function is a function that is a combination of one or more functions.
- 5. Read this as “ f of x is 5 if x is greater than 0 and less than 13 , 9 if x is greater than or equal to 13 and less than 55 , and 6.5 if x is greater than or equal to 55 . ” The rule for a piecewise function is different for different parts, (or pieces), of the domain (x-values) For instance, movie ticket prices are often different for different age groups. So the function for movie ticket prices would assign a different value (ticket price) for each domain interval (age group).
- 8. Restricting the domain of a function What is the domain? All real numbers What is the equation for this graph? f(x) = –2x – 5
- 10. What rule would you write for this piecewise function? Piecewise Functions x 2 – 3 if x ≥ –2
- 11. a) What is the value of y when x = –4? Give two ways to find it. Piecewise Functions b) Which equation would you use to find the value of y when x = 2? c) Which equation would you use to find the value of y when x = –2? x 2 – 3 if x ≥ –2
- 13. 2x + 1 if x ≤ 2 x 2 – 4 if x > 2 h(x) = Because –1 ≤ 2, use the rule for x ≤ 2 . Because 4 > 2, use the rule for x > 2. h(–1) = 2(–1) + 1 = –1 h(4) = 4 2 – 4 = 12 Evaluate the piecewise function for: x = –1 and x = 4.
- 14. 3x 2 + 1 if x < 0 5x – 2 if x ≥ 0 g(x) = Because –1 < 0, use the rule for x < 0. Because 3 ≥ 0, use the rule for x ≥ 0. g(3) = 5(3) – 2 = 13 g(–1) = 3(–1) 2 + 1 = 4 Evaluate each piecewise function for: x = –1 and x = 3
- 15. 12 if x < –3 20 if x ≥ 6 f(x) = Because –3 ≤ –1 < 6 , use the rule for – 3 ≤ x < 6 f(–1) = 15 Evaluate each piecewise function for: x = –1 and x = 3 15 if –3 ≤ x < 6 f(3) = 15 Because –3 ≤ 3 < 6 , use the rule for – 3 ≤ x < 6