Inverses & One-to-One
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The document discusses inverse functions and one-to-one functions. It provides examples of inverse functions, explains how to determine if a function is one-to-one, and how to find the inverse of a one-to-one function. It also describes properties of inverse functions, including that the domain of a function is the range of its inverse and their graphs are reflections across the line y=x.
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3) For a function to have an inverse, it must be one-to-one, meaning each input value maps to a unique output value. The horizontal line test determines if a function is one-to-one.
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The document defines inverse functions and provides examples. An inverse function f-1(x) undoes the original function f(x) so that f-1(f(x)) = x. For a function to have an inverse, it must be one-to-one meaning each output of f(x) corresponds to only one input x. The document gives examples of linear functions that are invertible and the function y=x2 that is not invertible because it is not one-to-one. It also states that if a function f(x) is one-to-one on its domain, then it has an inverse function and the domain of f(x) is equal to the range of the
4.1 Inverse Functions
Review composition of functions and one-to-one functions
Identify inverse functions
Write the equation of an inverse function
Inverse function
This document defines and provides examples of one-to-one, onto, and bijective functions. A one-to-one function maps distinct inputs to distinct outputs. An onto function maps the entire domain onto the range. A bijective function is both one-to-one and onto. The inverse of a bijective function is also discussed. Examples include functions from integers to integers and reals to reals.
11 the inverse trigonometric functions x
The document discusses one-to-one functions and inverse functions. A function f(x) is one-to-one if different inputs produce different outputs. The inverse of a one-to-one function f(x), denoted f^-1(x), is a function with the domain and range swapped such that f^-1(f(x)) = x and f(f^-1(x)) = x. While some inverse functions can be solved for algebraically, in general inverse functions must be determined through other means as algebraic solutions are not always possible.
C:\Fakepath\Stew Cal4e 1 6
The document discusses inverse functions and logarithms. It begins by introducing the concept of an inverse function using an example of bacteria population growth over time. It then defines inverse functions formally and discusses their key properties. The document explains that a function must be one-to-one to have an inverse function. It introduces the natural logarithm as the inverse of the exponential function with base e and discusses properties of logarithmic functions like logarithmic laws. Graphs of exponential, logarithmic and natural logarithmic functions are presented.
TYPES OF FUNCTION FOR JEE PREPARATION WITH EXAMPLES
TYPES OF FUNCTION FOR JEE PREPARATION WITH EXAMPLES
Functions 1 - Math Academy - JC H2 maths A levels
Slides for Functions lecture 1.
JC H2 Maths
A levels Singapore
www.mathacademy.sg
Copyright 2015 Math Academy
Introductory calculus
The document discusses relations and functions. A relation pairs elements from two sets, while a function pairs each element from the first set (domain) to only one element in the second set (range). An example is provided of a relation that is not a function because it pairs one element from the domain to two elements in the range. Functions can be represented numerically, algebraically, graphically, or verbally. Evaluating functions involves substituting values into the function formula. A function's domain consists of the valid inputs, while its range consists of the possible outputs.
CMSC 56 | Lecture 9: Functions Representations
Functions Representations
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 13, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
Folleto sobre las funciones
The document discusses several types of functions:
1) Linear functions have an expression that is a polynomial of first degree and their graph is a straight line. Examples are given.
2) Quadratic functions contain terms of up to second degree and their graph is a parabola. They can have one or multiple variables.
3) Cubic functions are polynomial functions of third degree with the form f(x)=ax3+bx2+cx+d. Their derivative is quadratic and integral is quartic.
4) Reciprocal or inverse functions satisfy the property that if f(x)=y then f-1(y)=x, undoing the mapping of the original function.
5) Ex
5.6 Function inverse. A handout.
Selected items from set theory and methodology and philosophy of mathematics and computer programming.
237654933 mathematics-t-form-6
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Functions
1) A function assigns each element of its domain to exactly one element of its codomain. The domain is the set of inputs and the codomain is the set of possible outputs.
2) A function can be represented by an equation, a mapping of inputs to outputs, or a graph. Properties of functions include being one-to-one, onto, bijective, increasing, decreasing.
3) The composition of two functions is the application of one function after the other. The inverse of a bijective function undoes the original mapping.
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Topic 3 Inverse of function, steps to find inverse and properties of inverse
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TYPES OF FUNCTION FOR JEE PREPARATION WITH EXAMPLES
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Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
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1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
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