# 3.1 Functions and function notation  (Page 7/21)

 Page 7 / 21

Using [link] , solve $\text{\hspace{0.17em}}f\left(x\right)=1.$

$x=0\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}x=2\text{\hspace{0.17em}}$

## 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] .

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 $\text{\hspace{0.17em}}q\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ both give output $\text{\hspace{0.17em}}n.\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ has a unique area measure given by $\text{\hspace{0.17em}}A=\pi {r}^{2},$ so for any input, $\text{\hspace{0.17em}}r,\text{\hspace{0.17em}}$ there is only one output, $A.$ The area is a function of radius $\text{\hspace{0.17em}}r.$

If the function is one-to-one, the output value, the area, must correspond to a unique input value, the radius. Any area measure $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ is given by the formula $\text{\hspace{0.17em}}A=\pi {r}^{2}.\text{\hspace{0.17em}}$ Because areas and radii are positive numbers, there is exactly one solution: $\sqrt{\frac{A}{\pi }}.$ So the area of a circle is a one-to-one function of the circle’s radius.

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.

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?
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.

## 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.

find the sum of 28th term of the AP 3+10+17+---------
I think you should say "28 terms" instead of "28th term"
Vedant
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
write down the polynomial function with root 1/3,2,-3 with solution
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
what is the answer to dividing negative index
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
give me the waec 2019 questions
the polar co-ordinate of the point (-1, -1)
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
tanh`(x-iy) =A+iB, find A and B
B=Ai-itan(hx-hiy)
Rukmini
Give me the reciprocal of even number
Aliyu
The reciprocal of an even number is a proper fraction
Jamilu
what is the addition of 101011 with 101010
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve