Question

PLEASE HELP ME! I WILL MEDAL AND FAN
What function is represented by the mapping diagram shown?

http://media.apexlearning.com/Images/201307/02/1fcd3bae-3d76-48f6-8be7-a30c03231c8b.JPG

Answer 1

A. F(x) = one half x
B. F(x) = x2
C. F(x) = x + 1
D. F(x) = 2x

Answer 2

The mapping diagram's bubble on the left is the independent variable (usually x) and the right bubble is the dependent variable (usually y, for F(x) in this case).
If you have made a choice, check your choice by evaluating F(x), using x from the left bubble, and see if F(x) matches the corresponding value in the right bubble. If all values match, you have the right answer.

For example, if F(x)=5x, take x=1 from the left bubble , and F(1)=5(1)=5 which does not match value of the right bubble (1), F(x) =5x is not the right answer.

Answer 3

so would the answer be \[F \left( x \right) = x^{2}\] @mathmate

Answer 4

want help

Answer 5

yes please

Answer 6

i'll help then

Answer 7

okay thanx

Answer 8

Find the inverse of the following functions:
a) f(x) = 2x + 4
y = 2x + 4 put y instead of f(x)
–2x = 4 – y
x = –2 + ½y We have divided through by –2

This is an equation where x is a function of y. The name of the variable doesn't matter so we can exchange the x and the y . If we call this function g we get the equation of the inverse function of f(x).

g(x) = y = ½x – 2.

We can check our result by putting in a number.
f(1) = 2·1 + 4 = 6 and g(6) = ½·6 – 2 = 3 – 2 = 1
If we use the general value a, we get
f(a) = 2a + 4 and g(2a+4) = ½·(2a+4) – 2 = a + 2 – 2 = a
b) f(x) = sin 2x, Df = [–/4,/4ñ , f(x) is increasing on this interval and therefore has an inverse function.
y = sin 2x
2x = sin–1 y
x = ½ sin–1 y
The inverse function is g(x) = ½ sin–1 x.
In most books the inverse of a function is written using the index –1 so that f(x) has the inverse function f –1(x). This does not mean 1/f, it's simply a notation for the inverse function.
In the above example f –1(x) = ½ sin–1 x.
Check.
f(/12) = sin 2/12 = ½
f –1(½) = ½·sin–1 ½ = ½·/6 = /12
c) f(x) = e2x
y = e2x
ln y = ln e2x = 2x
x = ½ ln y
The inverse function of f(x) is therefore f –1(x) = ½ ln x.
d) f(x) = x2 – 1, Df = R+
y = x2 – 1
x2 = y + 1


The inverse function is

e)


We choose Df = R+ and cube both sides of the equation

Then take the square root

The inverse function is



If the function f(x) is either always increasing or always decreasing then it has an inverse function f –1(x).

The Range of f(x) becomes the Domain of f –1(x).

We find the equation of the inverse by solving the equation y = f(x) for x.



Example 2

How do we need to limit the domain of the function f(x) = sin x so that it has an inverse function? We know that a continuous function that is always increasing ( or decreasing ) has an inverse.
So we look at the derivative of the function f(x) = sin x , f´(x) = cos x.


Using the unit circle we can see that cos x is positive from –/2 to /2 and negative on the rest of the circle.
Look at the graph of f(x) = sin x.


We see that the graph is increasing on –/2 < x < /2 so f(x) = sin x has an inverse if we limit the domain to this interval.
We could have chosen another interval, for example /2 < x < 3/2 where the function is decreasing but commonly the interval –/2 < x < /2 is chosen.

Example 3

Find the interval on which the function f(x) = x2 – 4x + 3 is increasing, limit the domain to this interval and then find the formula for the inverse function. Finally draw the graph of both f(x) and f–1(x) in the same coordinate system.
We begin by finding the vertex of the parabola by differentiating and finding where the derivative is zero. ( The tangent to f(x) at the vertex is horizontal and therefore the derivative is zero )

f(x) = x2 – 4x + 3
f´(x) = 2x – 4 = 0
2x = 4
x = 2
The vertex is where x = 2 and the function is increasing after that. We therefore choose the domain
Df = .
To find the equation of the inverse we need to solve the equation y = x2 – 4x + 3 for x.
y = x2 – 4x + 3

y – 3 = x2 – 4x

y – 3 + 4 = x2 – 4x + 4

y + 1 = (x – 2)2




Complete the square by adding half the coefficient of x (2 ) squared (4) to both sides of the equation and using the rule
a2 – 2ab + b2 = (a – b)2.


We choose the + as x is on the interval


Put in an x instead of the y


Now we draw the graph by first making a table of values.
Calculating f(2) tells us that f(x) = y takes values from –1 upwards ( the function is increasing ). We need therefore to start by finding f–1(–1).


The two graphs are shown below.


We note that the two graphs are mirror images of each other in the line y = x. (the line that bisects the angle between the x and y axes). We can also see this from the table of values. Each point on f(x) is the mirror image of a point on
f –1(x). For example (2, –1) is on f(x) and (–1, 2) is on f –1(x). (3, 0) is on f(x) and (0, 3) on f –1(x). In general if (a, b) is on one graph then (b, a) is on the graph of the inverse function.


The graph of a function and it's inverse function are always the mirror image of each other in the line y = x.

Example 4

Look at the function f(x) = ex and its inverse function g(x) = ln x.
The function f(x) = ex can take any value of x so its domain is all the real numbers R.
The values that f(x) = ex take are always positive so it's range is the interval .
The function g(x) = ln x can only take positive values of x , so it's domain is .
On the other hand the function g(x) = ln x gives all real number values so it's range is R.

The Range of a function is the same as the Domain of it's inverse and the Domain of a function is the Range of its inverse.

The graphs of the two functions are shown below. Notice that they are mirror images of each other in the line y = x .


Example 5

Find the interval on which the function is increasing. Choosing this interval as the domain find the equation of its inverse function then draw both graphs in the same coordinate system.

Differentiate, using the chain rule, to find where the function is increasing and where decreasing,





The denominator is always positive so its the x in the numerator that tells us where the function is increasing or decreasing.

The function is increasing for non negative values of x so we choose the domain

Df =
Next we solve for x to find the inverse.
y = 1 + (x2 + 1)½
y – 1 = (x2 + 1)½
(y – 1)2 = x2 + 1
x2 = (y – 1)2 – 1
= y2 – 2y + 1 – 1
= y2 – 2y

( choosing the positive value for x )

The inverse function is


The graphs of f(x), f –1(x) and the line y = x look like this.


Example 6

Find the inverse of the function and draw the graphs in the same coordinate system.

The graph has a vertical asymptote x = 1 and a horizontal asymptote y = 2. The domain does not contain x = 1.

We differentiate to find the slope of the graph.


The denominator is always positive and the numerator always negative which means that the slope of the graph is always negative. So the function is decreasing on all it's domain.

We find the inverse function by solving for x.

The inverse is

First get rid of the fractions then move all the terms with an x over to the left hand side of the equation.

Now x can be factorised out .

We see that y cannot be 2.



This function has a vertical asymptote in x = 2 and a horizontal asymptote in y = 1. This is the exact opposite to f(x) which has a vertical asymptote in x = 1 and a horizontal asymptote in y = 2.
Look at the two graphs.


The graphs and their asymptotes are mirror images of each other in the line y = x.

Answer 9

hope it helps

Answer 10

i know that's alot

Answer 11

but it can help

Answer 12

i guess

Answer 13

okay thanks so i think the answer is F(x)=\[x^{2}\]

Answer 14

could be but people said i can't give away answer

Answer 15

just like that

Answer 16

but like i said it could be the answer

Answer 17

omg who said that your just helping me i guessed what i think it is and all you doing is telling me if im right or not but okay

Answer 18

i did not understand that but okay

Answer 19

nvm but okay

Answer 20

what do you think is the answer in your multiple

Answer 21

choices sorry

Answer 22

i think the answer is B

Answer 23

if you think it's B. then go for it

Answer 24

i'm here to help people like you

Answer 25

OKAY!

Answer 26

if you help text if need

Answer 27

okay

Answer 28

ok

Answer 29

i got it right its B

Answer 30

i can help but not on weekends okay

Answer 31

i'm glad to help

Answer 32

OK

Answer 33

here i can friend you and give you a medal for being so kind :)

Answer 34

i'm mean fan you

Answer 35

aww thanks

Answer 36

your welcome

Answer 37

i did the same

Answer 38

what do you mean?

Answer 39

i fanned you and gave you a medal

Answer 40

ohhhhhhhhhhhhhhhhh thx!

Answer 41

like i said text if you want help okay

Answer 42

np and ok bye!

Answer 43

:)

Answer 44

see ya