For any real number A. B function f (x) satisfying f (a + b) + F (a-b) = 2F (a) * f (b), and f (0) is not equal to 0, it is proved that f (x) is an even function

For any real number A. B function f (x) satisfying f (a + b) + F (a-b) = 2F (a) * f (b), and f (0) is not equal to 0, it is proved that f (x) is an even function

It is proved that if we know that f (a + b) + F (a-b) = 2F (a) &; f (b), then,
Let a = b = 0, f (0) + F (0) = 2 [f (0)] ^ 2
If f (0) ≠ 0, f (0) = 1
Let a = 0, f (b) + F (- b) = 2F (0) f (b),
That is, f (x) = f (- x),
The function f (x) is even

Given that f (x + 1) is an even function, and f (x) monotonically decreases in the interval (1, + ∝), a = f (2), B = f (㏒ 3 & # 178;), C = f (1 / 2), then the order of a, B, C is?

F (x + 1) is even function
The translation of F (x) to the left by 1 unit gives f (x + 1)
X = 1 is the axis of symmetry of F (x)
∵ f (x) decreases monotonically on the interval (1, + ∝)
∝ f (x) increases monotonically on the interval (∝, 1)
a=f（2）,b=f（㏒3²）,c=f（1/2)
㏒ 3 & ෺ 178; nearest to x = 1
2 is farthest from x = 1
∴b>c>a
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Given the function f (x) = LG (1-x) / (1 + x), it is necessary to prove the process of F (x) + F (y) = f [(x + y) / (1 + XY)]!

f(x)+f(y)=lg(1-x)/(1+x)+lg(1-y)/(1+y)=lg(1-x)(1-y)/(1+x)(1+y)=lg(1+xy-x-y)/(1+xy+x+y)
f[(x+y)/(1+xy)]=lg{[(1-(x+y)/(1+xy)]/[(1+(x+y)/(1+xy)]=lg(1+xy-x-y)/(1+xy+x+y)=f(x)+f(y)

High school common English phrases, phrases,
For example, fond of

All(a.,pron.&n.)
All by oneself
Above all
After all
First of all
In all
Most of all
All at once
All of a sudden
All right, all right
All sorts of
All kinds of
All the best
All the more
All the same
All the year round
AS（conj.,adv.& pron.）
as… as… Same as
As a matter of fact
As a rule
As a result
As a whole
As if (as though)
As follows
As for
As (so) long as
As soon as
As soon as possible
As usual
As well
As well as
Might (may) as well
So as to
At(prep.)
At a time
At breakfast
At first
At home
At last
At least
At (the) most
At one time
At once
At night at night
At midnight
At present
At times
At sea
At one's own expense
At the bottom
At the end (of)
At the latest
At the mercy of Under the control of
At the head of Ahead of us
At the moment
At the same time
At work
Laugh at
Throw at Throw it away
Work hard at
By(prep.)
By accident
by air(sea,bus… ）By air (by boat, by bus...)
By chance
By day (night)
Day by day
By and by
By far
Learn (know) by heart
By mistake
By oneself
One by one
By the way
By turns
Side by side
By the side of nearby
Break(v.)
Break away from
Break down (a machine, a vehicle, etc) break down
Break forth
Break in break in break in break in break in
Break into
To break off suddenly
Break out
Break through
Break up, break up, break up
Bring(v.)
Bring about, bring about, bring about
To bring down; bring down
Bring forth
Bring in bring in bring in bring in bring in bring in bring in bring in bring in
Bring on
Bring out, explain, publish
Bring to an end
Bring up, bring up, bring up
Call(v.)
Call at (a place)
Call back
Call for to find (someone), like to pick up (something), want, demand, need to have
Call in call in call in call in
Call on (upon)
Call up call up call up call up call up call up
Come(v.)
Come about
Come across
Come along
Come down
Come into power (Office)
Come out
Come into being
Come to (a place), come to (an activity), talk about, regain consciousness
Come to a conclusion
Come to an end
Come to nothing (no good)
Do(v.)
Do a good job
Do away with
Do good, do good
Do harm, do harm
Do one a favor
Do one good for sb
Do one's best
Do the deed
Do one's duty
Do up, tidy up, repair, dress up, pack, tie, tie
Do with deal with
Do wonders
Do wrong (right)
Have sth (nothing) to do with
Down(adv.)
To bring down
Break down decomposition
Burn down Burn to the ground, burn to the ground
Get down
Hand down Pass it down
Put down
Tear down
Turn down
For(prep.)
For a while
For ever
For free
For the moment
Be fame for And famous
Have a gift for Have talent
Make a plan for Make plans
Stand for
In (one's) search for
As for
Care for
change… To use change
Fix a date for Date of
Give(v.)
Give a talk
Give birth to
Give in
Give off
Give one's life
Give sb a hand
Give one's greetings to
Give out
Give up give up give up give up
Get(v.)
Get about
Get along with
Get away, get away
Get away from
Get back
Get close to
Get down
Get down on one's knees
Get down to
Get into the habit of
Get hold of
Get in, get in, get back, get in, please
To take off (clothes, etc)
Get on, get on, get on, go on, go on
Get on well with
To overcome, endure, get rid of (a disease, etc)
Get rid of
Get round
Get through get through get through get through get through
Get together
Get up, get up, hold
Go(v.)
Go after
Go away
Go against
Go ahead, go ahead, go ahead
Go all out
Go bad
Go by
To engage in (a cause or activity)
Go on, go on, go on
Go off
Go out, go out
Review, examine, research
Go through
Go up, go up
In(prep.,adv.)
In a flash
In a word in a word
In a hurry
In (actual) fact
In a sense
In a short while
All in all
In any case
In battle
In case of (a situation)
In chains, in chains
In charge of
In common
In danger
In debt
In front
In front of front
In full text
In general
In memory of In order to Pay homage
In modern time
In one's opinion
In order to (that)
In other words
In peace
In public
In search of
In (one's) search for
In return
In silence
In short (= in a word)
In space
In spice of
Because, the reason is
In that case
In the air
In the charge In charge of
In the course of In the process
In the day
In the day time
In the future
In the end
In the meantime
In the middle (of) middle
In time
In turn
Believe in
Bring in
Call in
Hand in
Hand in hand
Drop in
Join in
Once in a while
Play a part in work
Stand in line
Succeeded in success
Take part in
Keep(v.)
Keep a promise
Keep a secret
Keep watch
Hold back (tears)
Keep body and soul together
Keep in mind
Keep off
Keep one's balance
Keep on
Keep out
Keep silent
Keep sb from doing sth
To maintain (an activity)
Keep up with
Look（v.）
Look after
Look as if
Look back (up)
Look down on (up on)
Look for
Look forward to
Look into
Look like
look on（upon）… Put sth regard as
Look out
Look out (for)
Look the same
Look through
Look over
Look up, look up, look up
Make（v）
Make clear
Make a decision
Make a promise
Make a plan for Make plans
Make a record
make friends （with）（