Question

The function f is defined by

Answer 1

\(f:x \rightarrow \frac{ x }{ x^2+1 },x \in\mathbb{R}\)

Answer 2

If \(a\in\mathbb{R}\) and \(a\neq0\),find the image of \(\frac{1}{a}\) under f.

Answer 3

Deduce that f is not one-to-one.

Answer 4

Show that if a,\(b\in\mathbb{R}\) with \(a>b\ge\mathbb{R}\),then f(b)>f(a).

Answer 5

Deduce that,if the domain of f is restricted to the subset of \(\mathbb{R}\) given by {x:\(x\ge~1\)},then f is one-to-one.

Answer 6

State the range of f in this case.

Answer 7

@Angle

Answer 8

i got the answer for f(1/a)

Answer 9

ewww math

Answer 10

f(1/a)= a/(1+a^2)

Answer 11

but i'm not sure how to do the rest.. xD

Answer 12

I really have a hard time with these problems
why can't they use plain english? ;-;

Answer 13

Haha..i wish they use plain english but It turns out d other way.. xD

Answer 14

can I... not try? x'D sorry

Answer 15

*crying*

Answer 16

it's okay..

Answer 17

if b < a
then f(b) > f(a)
because the bigger the number, the smaller it gets (because of how the x is squared in the denominator)

Answer 18

*the smaller the function gets

Answer 19

hmm,that makes sense..

Answer 20

so,i just need to show my working by choosing the smaller value for a
and bigger value for b..then,insert the values into the function..

Answer 21

for example
a=1/2 and b=1
sub. these value into the funtion
f(a)=a/(1+a^2)
f(1/2)=2/5
f(b)=a/(1+b^2)
f(1)=1/2

Answer 22

mhmm :)

Answer 23

alright!..
what about the next senctence..Deduce that,if the domain of f is restricted to the subset of \(\mathbb{R}\) given by {x:\(\x\ge~1)},then f is one-to-one.

Answer 24

{x:\(x\ge~1\)}

Answer 25

we just need to sketch the graph to show that the function is one-to-one,right?

Answer 26

@Zepdrix @Angle

Answer 27

i tried using desmos to sketch that graph

Answer 28

i think i managed to find the range of f...
but not sure how to do the second last sentence...

Answer 29

A function is one-to-one if it passes the Horizontal Line Test, which proves that there is one output to every input value. Reference source: mathwords

Answer 30

The original graph, unbounded by any parameters, would NOT be one-to-one

Answer 31

Your problem is telling you (the second last sentence) that if you "restrict the graph to domain \(\{x:x\ge1\}\)" then you need to tell "if [the new bounded graph] is one-to-one"

Answer 32

Here's a hint: does the bounded graph on domain \(\{x:x\ge1\}\) pass the Horizontal Line Test?

Answer 33

no,it didn't pass through the horizontal line test

Answer 34

i'm a little bit confused...
if the line touches the horizontal line test,does it considered as a one-to-one function? @kittybasil

Answer 35

for x > 1 it does pass the horizontal line test

Answer 36

@MARC
If your function graph touches a single horizontal line only once, then it passes the Horizontal Line Test.

Answer 37

got it...
Thanks for helping me... @Angle @kittybasil
^_^