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The document defines and provides information about common mathematical functions including linear, quadratic, square root, cubic, cube root, absolute value, greatest integer, rational, trigonometric, exponential growth and decay, and logarithmic functions. Tables are included that specify domains, ranges, x-intercepts, and y-intercepts for each function.
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Parent Function Project
The document defines and provides information about common mathematical functions including linear, quadratic, square root, cubic, cube root, absolute value, greatest integer, rational, trigonometric, exponential growth and decay, and logarithmic functions. Tables are included that specify domains, ranges, x-intercepts, and y-intercepts for each function.
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An asymptote is an imaginary line that a graph approaches but never touches as it goes to infinity. The document provides examples of graphs with rational functions that have vertical and horizontal asymptotes. It also discusses the square root function and how it is the inverse of the quadratic function. The square root function is only defined for non-negative inputs. Changing the parameters a and b in the square root function changes the shape and orientation of the graph.
Sums.of.equivalent.sequences.of.positive.operators.Taft.Beamer.Presentation
The document discusses spaces and operators in finite dimensions. It provides examples of 0, 1, 2, and 3-dimensional spaces. Operators are defined as transformations that preserve certain properties of a space. Examples of 1D, 2D and 3D operators are shown as stretches, projections and rotations. Operators can be represented by matrices, which depend on the coordinate system chosen. Operators can be added and multiplied by numbers by performing the corresponding operations on their matrix entries.
Functions
This document discusses functions and how to determine if a relation is a function. It defines a function as a relation where each element of the domain maps to exactly one element of the range. It provides examples of tables, coordinate points, and graphs, and explains how to use the vertical line test to determine if a graph represents a function. It also discusses how to determine if a relation is a function algebraically by solving for the output variable and checking if there is only one output for each input.
QUESTION 1Find the domain of the function Domain all real.docx
QUESTION 1
Find the domain of the function
Domain: all real numbers x except x=6
Domain: all real numbers x except x=8
Domain: all real numbers x except x = -8
Domain: all real numbers x
Domain: all real numbers x except x = -6
5 points
QUESTION 2
Find the domain of the function
Domain: all real numbers x except x = 7
Domain: all real numbers x except x = ±49
Domain: all real numbers x except x = ±8
Domain: all real numbers x except x = -7
Domain: all real numbers x except x = ±7
5 points
QUESTION 3
Determine the domains of f and g.
Domain of f: all real numbers x except x = -6
Domain of g: all real numbers x
Domain of f: all real numbers x except x = ±6
Domain of g: all real numbers x
Domain of f: all real numbers x except x = ±36
Domain of g: all real numbers x except x=6
Domain of f: all real numbers x except x = 6
Domain of g: all real numbers x
Domain of f: all real numbers x except x = ±6
Domain of g: all real numbers x except x=6
5 points
QUESTION 4
Find the domain of
all real numbers except x = -9
all real numbers except x = 81
all real numbers except x = 9
all real numbers except x = 9 and x = -9
all real numbers
5 points
QUESTION 5
Find the domain of the function and identify any vertical and horizontal asymptotes.
Domain: all real numbers x except x = 2
Vertical asymptote: x = 2
Horizontal asymptote: y = -1
Domain: all real numbers x
Vertical asymptote: x = -2
Horizontal asymptote: y = 0
Domain: all real numbers x except x = 0
Vertical asymptote: x = 0
Horizontal asymptote: y = -1
Domain: all real numbers x except x = 2
Vertical asymptote: x = 0
Horizontal asympotote: y = -2
Domain: all real numbers x except x = -2
Vertical asympotote: x = 0
Horizontal asympotote: y = 2
5 points
QUESTION 6
Find the domain of the function and identify any vertical and horizontal asympototes.
Domain: all real numbers x except x = 0
Vertical asympotote: x = 0
Horizontal asympotote: y = -2
Domain: all real numbers x except x = -6
Vertical asympotote: x = -6
Horizontal asympotote: No horizontal asymptote
Domain: all real numbers x except x = 6
Vertical asympotote: x = 6
Horizontal asympotote: No horizontal asympotote
Domain: all real numbers x except x = 6
Vertical asympotote: x = 6
Horizontal asymptote: y = 2
Domain: all real numbers x
Vertical asymptote: x = -6
Horizontal asymptote: y = 0
5 points
QUESTION 7
Find the domain of the function and identify any vertical and horizontal asympototes.
Domain: all real numbers x
Vertical asympotote: No vertical asymptote
Horizontal asymptote: y = 9
Domain: all real numbers x except x = 0
Vertical asymptote: x = 0
Horizontal asymptote: y = -8
Domain: all real numbers x
Vertical asymptote: x = -9
Horizontal asymptote: y = 0
Domain: all real numbers x except x = 9
Vertical asymptote: No vertical asymptote
Horizontal asymptote: No horizontal asymptote
Domain: all real numbers x except x = 9
Vertical asymptote: x = 9
Hori ...
Rational Functions
This document discusses rational functions and their properties. It defines a rational function as a function of the form f(x) = p(x)/q(x) where p and q are polynomials. It then discusses how to find the domain, vertical asymptotes, horizontal asymptotes, and oblique asymptotes of rational functions. The key points are: 1) the domain excludes values where the denominator equals 0, 2) vertical asymptotes occur where the denominator equals 0, 3) the degree of the numerator and denominator determine if there is a horizontal or oblique asymptote. Comparing degrees is essential to finding asymptotes of rational functions.
Function evaluation, termination, vertical line test etc
The document discusses functions and their inverses. It defines key terms like domain, range, and inverse functions. It provides examples of determining if a relationship is a function using the vertical line test or one-to-one mapping. The document also covers finding inverses of straight lines, parabolas, exponential graphs and logarithmic graphs. It shows how to sketch these graphs and determine their domains and ranges. Finally, it discusses solving cubic expressions through finding factors and remainders.
Optimization Techniques.pdf
This document discusses optimization techniques and provides examples to illustrate key concepts in optimization problems. It defines optimization as finding extreme states like minimum/maximum and discusses how it is applied in various fields. It then covers basic definitions like design variables, objective functions, constraints, convexity, local vs global optima. Examples are given to show unconstrained vs constrained problems and illustrate active, inactive and violated constraints. Optimization techniques largely depend on calculus concepts like derivatives and hessian matrix.
Inverse Function.pptx
This document provides information about inverse functions including:
- The inverse of a function is formed by reversing the coordinates of each ordered pair. A function has an inverse only if it is one-to-one.
- The domain of the inverse function is the range of the original function, and the range of the inverse is the domain of the original.
- To find the inverse of a one-to-one function, replace f(x) with y, interchange x and y, then solve for y in terms of x and replace y with f^-1(x).
- Several examples are provided to demonstrate finding the inverse of different one-to-one functions by following the given steps.
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.
2nd Test - Parent Functions
The document discusses parent functions and provides examples of common parent functions including constant, linear, quadratic, absolute value, and exponential functions. It defines a parent function as the basic form of a function type and explains that parent functions allow us to determine traits of related functions. The document then lists the domain, range, and interval notation for each parent function type. It concludes by providing examples of functions and asking the reader to identify their parent function and range.
Higher Maths 1.2.1 - Sets and Functions
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.
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QUESTION 1Find the domain of the function Domain all real.docx
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Function evaluation, termination, vertical line test etc
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More from Joanne Rosa Crooks
Vertical stretch & shrink
This document discusses transformations of the square root function y=√x. It includes:
1) Matching equations like y=3√x and y=√x/2 to their graphs by graphing the parent function first.
2) Explaining that a negative sign in front of the square root, like y=-√x, reflects the graph over the x-axis.
3) Having students work in groups to draw transformed square root graphs, identify the transformation, and write the domain and range.
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The document discusses adding and subtracting radicals. It reviews collecting like terms and then explains that to add or subtract radicals, you add the coefficients of like terms, which are radicals that have the same index and radicand. Examples are provided to demonstrate adding and subtracting radicals.
Rubric for Graphing parent functions project
1. The project requires students to graph the 13 parent functions and apply transformations to create child functions.
2. Students must complete a parent function foldable with information on all 13 functions and create a poster showing the graphs of each parent function and one example of a child function using a transformation.
3. The poster will be graded based on neatness, completeness of information and transformations, and visual appeal.
Right triangles day2
1. The document discusses trigonometric ratios and how to use them to solve for missing side lengths and angle measures in right triangles.
2. It provides examples of setting up trig ratios, using the Pythagorean theorem, and using inverse trig functions to find missing angles.
3. The key steps are to label the sides of the right triangle, set up the appropriate trig ratios based on which information is known or missing, and use trig identities or the inverse functions to calculate the missing information.
Right triangles day1
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Day1 1st
This one sentence document simply states the date, May 12, 2014, three times without providing any additional context or information.
U8 day1 4th
This document discusses the parts of a right triangle, listing the opposite leg, adjacent leg, and hypotenuse multiple times on May 4, 2014. It focuses on the basic geometric terms for the sides of a right triangle.
Common Logarithms
This document discusses common logarithms and how to evaluate logarithmic expressions with and without a calculator. It provides examples of rewriting exponential expressions as logarithmic expressions by setting them equal to variables and manipulating the equations. It also introduces the change of base formula for evaluating logarithms with bases other than 10.
U6 d2 word problems hw
This document contains 7 word problems about exponential growth and decay models. The problems cover topics like population growth, healthcare costs, radioactive decay, savings accounts, milk consumption, population of Washington D.C., and guppy population growth. For each problem, the student is asked to write an exponential function model, make predictions based on the model, or calculate other related values. The overall goal is to practice applying exponential functions to real-world scenarios involving growth and decay over time.
Day2 3rd & 4th
This document appears to be a log of activities that took place over two days, April 3rd and 4th, 2014. However, no specific activities or events are described within the document itself, which only repeats the date header five times without providing any additional context or information about what occurred.
U6 day2 1st pd.
This 3 sentence summary provides the high level information from the document. The document appears to be notes from a class titled "U6 day2 1st pd." that was held on April 22, 2014. It includes the title and date repeated 3 times with no other context or details provided.
Parent graphs cubic & cube root
This document discusses parent graphs of cubic and cube root functions. Cubic functions have a graph that is shaped like a curve opening upwards or downwards, while cube root functions have a graph that curves smoothly from left to right. Both cubic and cube root graphs pass the vertical line test and are one-to-one functions.
Parent graphs cube root & cubic
The document discusses parent graphs of cube root and cubic functions. It explains that the parent graph of the cube root function y=√x is a V-shape that opens to the right, and the parent graph of the cubic function y=x^3 is shaped like an upside-down V that opens downward.
4th period more graphing polynomials
This document discusses graphing polynomials and provides examples. It lists the degree, whether the polynomial is odd or even, the number of turns, whether coefficient a is positive or negative, the factored form, and the zeroes of two polynomials. The first polynomial has zeroes of -1 and 5, and the second has zeroes of 0, -3, and 2.
Factoring Sum and Difference of Cubes
The document contains notes from Lesson 2 that were taught in all 3 classes on March 20, 2014. The notes were repeated multiple times throughout.
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Graphing parent functions project1
- 2. FoldHere Glue this side to your notebook. Glue this side to your notebook. Linear Function Quadratic Function Square Root Function Cubic Function
- 3. FoldHere Glue this side to your notebook. Glue this side to your notebook. Cube Root Function Absolute Value Function Greatest Integer Function Rational Function
- 4. FoldHere Glue this side to your notebook. Glue this side to your notebook. Exponential Growth & Logarithm Function Exponential Decay & Logarithm Function
- 5. x y -3 -2 -1 0 1 2 x y -3 -2 -1 0 1 2 x y -3 -2 -1 0 1 2 x y -3 -2 -1 0 1 2 Linear 𝑓( 𝑥) = 𝑥 Domain: Range: x-int: y-int: x y x y x y x y Domain: Range: x-int: y-int: Domain: Range: x-int: y-int: Domain: Range: x-int: y-int: Quadratic Square Root Cubic 𝑓( 𝑥) = 𝑥! 𝑓( 𝑥) = √ 𝑥 𝑓( 𝑥) = 𝑥!
- 6. x y -3 -2 -1 0 1 2 x y -3 -2 -1 0 1 2 x y -3 -2 -1 0 1 2 x y -3 -2 -1 0 1 2 x y x y x y x y Domain: Range: x-int: y-int: Cube Root 𝑓( 𝑥) = √ 𝑥 ! Domain: Range: x-int: y-int: Domain: Range: x-int: y-int: Domain: Range: x-int: y-int: Absolute Value Rational Greatest Integer 𝑓( 𝑥) = ⌊ 𝑥⌋ 𝑓( 𝑥) = | 𝑥| 𝑓( 𝑥) = 1 𝑥
- 7. FoldHere Glue this side to your notebook. Glue this side to your notebook. Sine Function Cosine Function Tangent Function TRIG Functions
- 8. x y -3 -2 -1 0 1 2 x y -3 -2 -1 0 1 2 Exponential Growth & Logarithm Function 𝑓( 𝑥) = 𝑏! 𝑊ℎ𝑒𝑟𝑒 |𝑏| > 0 Exponential Growth & Logarithm Function 𝑓( 𝑥) = 𝑏! 𝑊ℎ𝑒𝑟𝑒 0 < |𝑏| < 1 2 Domain: Range: x-int: y-int: Domain: Range: x-int: y-int: x y x y
- 9. x y 0 90 180 270 360 x y 0 90 180 270 360 x y 0 90 180 270 360 Sine Function Cosine Function Tangent Function 𝑓( 𝑥) = sin 𝑥 𝑓( 𝑥) = cos 𝑥 𝑓( 𝑥) = tan 𝑥 Domain: Range: x-int: y-int: Domain: Range: x-int: y-int: Domain: Range: x-int: y-int: