One-to-One Functions
* One-to-one function is where each x-value corresponds to one y-value, and each y-value corresponds to only one x-value *
* Horizontal line test – a function is one-to-one if every horizontal line intersects the graph of the function at most once * Examples: Determine whether the following functions are one-to-one

* In general, a function that is either increasing or decreasing on its entire domain, such as must be one-to-one * Tests to Determine Whether a Function is One-to-One
* Show that f(a) = f(b) implies a = b; then the function is one-to-one * Every y-value corresponds to no more than one x-value; to show that the function is not one-to-one, find at least two x-values that produce the same y-value * Use the horizontal line test

* If the function either increases or decreases on its entire domain, then it is one-to-one

Inverse Functions
* Let be a one-to-one function. Then g is the inverse function of if: *
* Examples: Determine whether f and g are inverses *
*
*
* Inverse of a function is denoted as
* By definition of inverse function, the domain of f is the range of and the range of f is the domain of. * To find the inverse of a one-to-one function, interchange the x- and y-values of each of its ordered pairs. * Functions that are not one-to-one do not have an inverse

* Examples: If the functions are one-to-one, find their inverses. *

* Steps for Finding the Equation of the Inverse of a Function * Interchange x and y
* Solve for y
* Replace y with
* Examples: For each function that is one-to-one, a) write an equation for the inverse function in the form , b) graph the function & its inverse, c) give the domain and range of the function...

...Running Header: INVERSE 1
Composition and Inverse
Esther Bakeberg
MAT222: Week 5 Assignment
Donna Wall
July 21, 2014
INVERSE 2
Composition and InverseFunctions give us an opportunity for using expressions with different values. The
values can help business owners, (small or large), data collectors and analysts, and even the
consumer compare rates and data. Functions also extend independent (x) and dependent
(y) variables by graphing in the coordinate plane and creating a visual demonstration of
the relationship.
The functions I will be using in the required problems are
f(x) = 5x – 3 g(x) = x2 +2 h(x) = 3 +x / 7
The first task is to compute (f – h) (4).
(f – h) (4) = f(4) – (h - 4) The rules of composition allow each function to be
calculated separately and then subtracted.
f(4) = 5(4) – 3 The x was replaced with the 4 from the problem.
f(4) = 20 – 3...

...Composition and Inverse
Jack Lewis
Mat 222
Instructor: Dr. Dariush Azimi
January 14, 2013
Composition and Inverse
The assignment paper for this week is to define the following functions of Elementary and Intermediate Algebra: Problem number 1) fx=2x+5, Problem number 2) gx=x2- 3, and Problem number 3) hx= 7-x3.
Compute (f – h)(4).
Evaluate the following two compositions: and .
Transform the g(x) function so that the graph is moved 6 units to the right and 7 units down.
Find the inversefunctions and .
1. Write a two to three page paper that is formatted in APA style and according to the Math Writing Guide. Format your math work as shown in the example and be concise in your reasoning. In the body of your essay, please make sure to include:
* Your solutions to the above problems, making sure to include all mathematical work for both problems, as well as explaining each step.
* A discussion of the applicability of functions to the real world, based upon your reading of Chapter 11 of Elementary and Intermediate Algebra. Be sure to use specific examples, a brief discussion of why your examples are important, and to cite your sources.
The paper must be at least two pages in length and formatted according to APA style. Cite your resources in text and on the reference page. For information regarding APA samples and tutorials, visit the...

...
COMPOSITION AND INVERSEFUNCTIONS
Composition and InverseFunctions
Kimberly Harris
MAT 222 Week 5 Assignment
Instructor: Donna Wall
July 18, 2014
Composition and InverseFunctions
In this week’s assignment I am given three Composition and InverseFunctions. Functions gives an opportunity for manipulating experiences using different values. What these values does is to help business owners and others the opportunity to compare rates and dates. Functions can extend independent (x) and dependent (y) variables by graphing the coordinate plane and to create a visual demonstration of the relationship.
The three functions that will be used in the following problems are as follows:
f(x) = 2x = 5 g(x)= x² – 3 h(x) = 2 – x
3
The first thing I have to do is to compute (f – h)(4).
(f – h)(4) = f(4) – h(4) Because of the rules of composition, each function can be calculated separately and then subtracted....

...
Composition and Inverse
Student Name
MATH 222 Week 5 Assignment
Instructor Name
Date
Composition and Inverse
This week assignment was to understand and solve for composition and inversion. With the given problems and functions I will demonstrate how to compute a problem using composition. Then finally I will demonstrate how to solve a function using inversion.
I will define the following functions to solve the problem.
f(x)+2x+5 g(x)=x^2-3 h(x)=7-x
______
3
The first step is to compute the required problem (f-h)(4).
(f-h)(4)
f(4)-h(4) The first step is to multiply 4 into f and h to be subtracted from each other. Since the equation require functions f and h, we will plug them into the problem.
f(4)-h(4)= [2(4)+5]-[7-(4)]
_____
[3] Solve for functions f and h while substituting the x for the number 4
=13 - 11
___
3 Subtract
= 28
___
3 Final answer
The next step is to evaluate the following compositions.
A: (f o g)(x)= 2(x^2-3)+5 Multiply 2 to the equation in the parenthesis
= 2x^2-6+5 Sutract -6 from 5
= 2x^2-1...

...the family. Hence,
i) determine the domain, codomain, objects, images and range.
ii) recognise the type of relation.
iii) represent each relation above using other methods.
c) Based on (b), which relation is a function? State your reason.
Part 2
a) You are required to come out with an attractive and creative card. Your card must have the following information.
i) a family photo
ii) a family tree
b) Write a short description about your family in not more than 150 words.
Part 3
Based on Diagram 3, answer the following questions.
Diagram 3
Find
a) function [pic] and function [pic].
b) the inversefunction [pic] and [pic].
c) composite function [pic], [pic], [pic] and [pic].
d) function [pic] and [pic]. Is [pic] = [pic]?
e) What can you deduce from the following composite function?
[pic]and [pic]
Hence, deduce the value of [pic].
Further Exploration
a) The function g is defined by [pic]. Find
i) [pic], [pic], [pic], [pic], [pic] and [pic].
ii) Hence, state the function [pic] and [pic], where n is a positive integer.
b) Given the function [pic]is defined by [pic]. Find [pic]. Therefore deduce the inversefunctions for the followings. Check your answers using...

...SEK MEN SAINS MUZAFFAR SYAH MELAKA 75450 AYER KEROH MELAKA
Teaching
&
learning
Additional mathematics
Form 4
NAME:…………………………………………… FORM :…………………………………………..
Date received
……………………………………..
Date completed …………………………. Marks of the Topical Test : ……………………………..
Prepared by : PN HJH SARIPAH AHMAD Sekolah Menengah Sains Muzaffar Syah Melaka 75450 AYER KEROH MELAKA
1
LEARNING AREA : FUNCTIONS Learning Objectives : Understand the concept of relations Learning Outcomes : Student will be able to
1.1 Represent relation using a) arrow diagram , b) ordered pairs, c)graphs 1.2 Identify domain, co domain, object, image and range of a relation.
1.3 Classify a relation shown on the mapped diagram as : one to one many to one , one to many or many to many
1.1a Representing a relation between two sets by using an arrow diagram Example 1 Suppose we have set A = { 0,1,2,3 } and set B = { 0,1,2,3,4,5,6} .Let us examine the relations “is one less than “ from set A to Set B . One way to show the relations is to draw an arrow diagram as shown below . The arrows relate the elements in A to the elements in B
A
is one less than
B
•6 •5
In the space below give other example to show a relation
3 2 1 0 .
FIG 1.1
4 3 2 . 1 •0
A relation from set A to set B is an association of elements of A with elements of B . Exercise 1 1 Two set of numbers, P and Q are shown below Complete the arrow diagram to show the relations “ is grater...

...CIRCULAR FUNCTIONS
A different name of an angle is circular functions. Communicate the direction of a triangle to the length of the surface of a triangle. Trigonometric functions are important of triangles and form episodic occurrence, along with many complementary applications. Trigonometric functions have a wide range of uses including calculating indefinite lengths along with angles in triangles.
Trigonometricfunctions are normally specific as ratios of two sides of a right triangle including the angle, and able to equally specific as the lengths of different line segments from a unit circle.
More modern significance communicate them an infinite series or as solutions of specific different equations, allowing their extension to subjective positive also negative values and complex numbers. The sine with cosine functions are with usually used to model periodic function. Circular functions along angle θ are:
Sine Function:
sin θ = OppositeHypotenuse
Cosine Function:
cos θ = AdjacentHypotenuse
Tangent Function:
tan θ = OppositeAdjacent
Cosecant Function:
csc θ = HypotenuseOpposite = 1sinθ
Secant Function:
sec θ = HypotenuseAdjacent = 1cosθ
Cotangent Function:
cot θ = AdjacentOpposite = 1tanθ
A function of a direction in a right-angled triangle to be specific the...

...
Composition and Inverse
Cindy Dean
Intermediate Algebra: MAT 222
Instructor:
February 14, 2014
Composition and Inverse
In this week’s assignment, I will be solving functions with different values and variables. Many companies and businesses, use these methods to either make progress or to change something that will benefit their success. The first function is:
(f – h)(4)
f(4) – h(4) I multiplied 4 with each variable.
f(4) = 2(4) + 5 The x is replaced with 4.
f(4) = 13 I used the order of operation to evaluate this
function.
h(4) = (7 – 3)/3 I will repeat the steps that I used in the previous
problem for the variable, h.
h(4) = 3/3 I used the order of operation to simplify.
h(4) = 1 This is the solution for h.
(f – h)(4) = 13 + 1 I combined the values of f and h.
(f – h)(4) = 14 This is my solution.
The next problem, I will be finding the solution for g(x). I will replace the x in the f function with the g function. My function will look like this:
(f° g)(x) = f(g(x))
(f° g)(x) = f (x^2 – 3) My g function is replacing the x.
(f° g)(x) = 2(x^2 – 3) + 5
(f° g)(x) = 2x^2 – 6 + 5 I used the order of operation to simplify
my answer.
(fg)(x) = 2x^2 – 1 This is my solution.
This problem, I will be composing for (hg)(x).
(h° g)(x) =...