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Using [link] , solve f ( x ) = 1.

x = 0 or x = 2

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Determining whether a function is one-to-one

Some functions have a given output value that corresponds to two or more input values. For example, in the stock chart shown in [link] at the beginning of this chapter, the stock price was $1000 on five different dates, meaning that there were five different input values that all resulted in the same output value of $1000.

However, some functions have only one input value for each output value, as well as having only one output for each input. We call these functions one-to-one functions. As an example, consider a school that uses only letter grades and decimal equivalents, as listed in [link] .

Letter grade Grade point average
A 4.0
B 3.0
C 2.0
D 1.0

This grading system represents a one-to-one function because each letter input yields one particular grade-point average output and each grade-point average corresponds to one input letter.

To visualize this concept, let’s look again at the two simple functions sketched in [link] (a) and [link] (b) . The function in part (a) shows a relationship that is not a one-to-one function because inputs q and r both give output n . The function in part (b) shows a relationship that is a one-to-one function because each input is associated with a single output.

One-to-one function

A one-to-one function    is a function in which each output value corresponds to exactly one input value. There are no repeated x - or y -values.

Determining whether a relationship is a one-to-one function

Is the area of a circle a function of its radius? If yes, is the function one-to-one?

A circle of radius r has a unique area measure given by A = π r 2 , so for any input, r , there is only one output, A . The area is a function of radius r .

If the function is one-to-one, the output value, the area, must correspond to a unique input value, the radius. Any area measure A is given by the formula A = π r 2 . Because areas and radii are positive numbers, there is exactly one solution: A π . So the area of a circle is a one-to-one function of the circle’s radius.

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  1. Is a balance a function of the bank account number?
  2. Is a bank account number a function of the balance?
  3. Is a balance a one-to-one function of the bank account number?

a. yes, because each bank account has a single balance at any given time; b. no, because several bank account numbers may have the same balance; c. no, because the same output may correspond to more than one input.

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  1. If each percent grade earned in a course translates to one letter grade, is the letter grade a function of the percent grade?
  2. If so, is the function one-to-one?
  1. Yes, letter grade is a function of percent grade;
  2. No, it is not one-to-one. There are 100 different percent numbers we could get but only about five possible letter grades, so there cannot be only one percent number that corresponds to each letter grade.
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Using the vertical line test

As we have seen in some examples above, we can represent a function using a graph. Graphs display a great many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. By convention, graphs are typically constructed with the input values along the horizontal axis and the output values along the vertical axis.

Questions & Answers

bsc F. y algebra and trigonometry pepper 2
Aditi Reply
given that x= 3/5 find sin 3x
Adamu Reply
4
DB
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Joeval Reply
-16a+6b+2c
Will
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Joeval
(x2-2x+8)-4(x2-3x+5)
Ayush Reply
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Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
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master
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master
Y
master
X2-2X+8-4X2+12X-20=0 (X2-4X2)+(-2X+12X)+(-20+8)= 0 -3X2+10X-12=0 3X2-10X+12=0 Use quadratic formula To find the answer answer (5±Root11i)/3
master
Soo sorry (5±Root11* i)/3
master
x2-2x+8-4x2+12x-20 x2-4x2-2x+12x+8-20 -3x2+10x-12 now you can find the answer using quadratic
Mukhtar
explain and give four example of hyperbolic function
Lukman Reply
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
Racelle Reply
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Anurag Reply
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
Jhovie Reply
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
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Jhovie
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Charmaine Reply
A banana.
Yaona
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Mark Reply
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Miranda
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Miranda
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jai
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Miranda
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Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
sita Reply
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Propessor
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Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
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jai
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jai
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Propessor
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jai
What is algebra
Pearl Reply
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
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Jeffrey
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Miranda
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Jeffrey
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Miranda
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Jeffrey
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Miranda
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Miranda
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Jeffrey
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Jeffrey
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Miranda
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Miranda
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Steve
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Steve
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Jeffrey
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Jeffrey
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Miranda
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Nelson Reply
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Miranda
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Jeffrey
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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