专题2.2函数的性质：单调性、奇偶性、对称性与周期性十二大题型（原卷版）_11.4-2_2026-11-04

      
      
专题2.2 函数的性质：单调性、奇偶性、对称性与周期性【十二大题型】

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é 1ê2023·ëìëí·ßîïðñ
𝑓(𝑥) = 𝑥2−4|𝑥| + 3" ò¡&'wê    ñ
AŽ(−∞,−2)
BŽ(−∞,−2)_(0,2)
CŽ(−2,2)
DŽ(−2,0)_(2, + ∞)
°8 1-1ê2024·óô·xßñõ
𝑓(𝑥)~𝑅¾ÏŸ
U¤ö÷dáx´¨"w(øø)
AŽ𝑦=
1
𝑓(𝑥)~𝑅¾Ï¡
BŽ𝑦= |𝑓(𝑥)|~𝑅¾ÏŸ
CŽ𝑦= −
1
𝑓(𝑥)~𝑅¾ÏŸ
DŽ𝑦= −𝑓(𝑥)~𝑅¾Ï¡
°8 1-2ê2024·ùú·µßñ56
𝑓(𝑥) = 𝑥2−2,𝑥≥0,
𝑥+ 3,𝑥< 0,
 
â𝑓(𝑎) = 𝑓(𝑎+ 3)U¤𝑔(𝑥) = 𝑎𝑥2 +𝑥" 
òŸ&'Ïê    ñ
AŽ 1
8 , + ∞
BŽ −∞, 1
8
      
      
CŽ 1
2 , + ∞
DŽ −∞, 1
2
°8 1-3ê2024·ùûûü·ßîïðñ56c~&'(−𝑚,𝑚)(𝑚> 0)¾U/‘ÏR"
𝑓(𝑥)ãä
①
ý0 < 𝑥< 𝑚…U𝑓(𝑥) > 0”②:»c‘|þb"`
 a9b ÿãä
𝑓(𝑎+ 𝑏) =
𝑓(𝑎)+𝑓(𝑏)
1−𝑓(𝑎)𝑓(𝑏)Ž¤ê    ñ
AŽ𝑓(0) = 1
BŽ∀𝑥1,𝑥2,−𝑚< 𝑥1 < 𝑥2 < 𝑚,𝑓(𝑥1) > 𝑓(𝑥2)
CŽ
𝑓(𝑥)~&'(0,𝑚)¾ ò¡
DŽ
𝑓(𝑥)~&'(−𝑚,𝑚)¾ òŸ
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é 2ê2024·óôĀā·µßñ56
𝑓(𝑥) = −𝑥2 +𝑎𝑥+ 1~(2,6)¾D U¤𝑎"Ô/ĂăÏê    ñ
AŽ(2,6)
BŽ(−∞,2] ∪[6, + ∞)
CŽ(4,12)
DŽ(−∞,4] ∪[12, + ∞)
°8 2-1ê2023·ĄąĆć·xßñõ𝑎∈𝑅U¤“𝑎> −2”w“
𝑓(𝑥) = 2𝑥2 +4𝑎𝑥+ 1~(2, + ∞)¾ ò
Ÿ”"ê    ñ
AŽ†QD{L¶·
BŽ{LD†Q¶·
CŽ†L¶·
DŽĈD†QĉD{L¶·
°8 2-2ê2023·ĊúċČ·xßñ56
𝑓(𝑥) =
−𝑥2 + 2𝑎𝑥,𝑥≤1
(3−𝑎)𝑥+ 2,𝑥> 1
 
wc~𝑅¾"Ÿ
U¤𝑎"Ô/
Ăăwê    ñ
AŽ[1,3)
BŽ[1,2]
CŽ[2,3)
DŽ(0,3)
°8 2-3ê2023·ćčĎď·xßñ56
𝑓(𝑥)"c‘Ï𝑅UĐ~Ý
𝑡(𝑡> 0)U±Ö:þb𝑥∈𝑅Uđ
Æ𝑓(𝑥+ 𝑡) = 𝑓(𝑥)Uý𝑥∈[0,𝑡)…U𝑓(𝑥) = |𝑥−𝑡
2|Žâ𝑓(𝑥)~&'(3,4)¾ ò¡U¤ t ".C/Ïê    ñ
AŽ3
BŽ
8
3
CŽ2
DŽ
8
5
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é 3ê2024·ùú¾Ē·xßñ.
 f(x)ēĔxĕ
1
𝑥~[−2,−
1
3]¾".B/wê    ñ
AŽ 
3
2 
BŽĔ
8
3
CŽĔ2
DŽ2
°8 3-1ê2024·ĖėĘć·µßñý`
𝑡°…U
𝑓(𝑥) = |𝑥2 + 𝑡|,𝑥∈[−4,4].B/".C/Ï
ê    ñ
AŽ2
BŽ4
CŽ6
DŽ8
      
      
°8 3-2ê2024·ęôĚĖ·ßîïðñ56
𝑓(𝑥) = 𝑥+
4𝑎
𝑥+𝑏,𝑥∈[𝑏, + ∞)U®Á𝑏> 0,𝑎∈𝑅Uě𝑀Ï𝑓(𝑥)
".C/U¤ý𝑀= 2…U𝑎"Ô/ĂăÏê    ñ
AŽ𝑎>
1
3
BŽ𝑎<
1
3
CŽ𝑎>
1
4
DŽ𝑎<
1
4
°8 3-3ê2024·ćčĜc·µßñ56
𝑓(𝑥) = {1−|𝑥+ 1|,𝑥< 0
𝑥2−2𝑥,𝑥≥0 Uâ`
𝑚∈[−2,0]U¤|𝑓(𝑥)−𝑓(−1)|
~&'[𝑚,𝑚+ 2]¾".B/"Ô/Ăăwê    ñ
AŽ[1,4]
BŽ[2,4]
CŽ[1,3]
DŽ[1,2]
 4  
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é 4ê2024·ĝĞ·ßîïðñ56
𝑓(𝑥)"c‘Ï𝑅U𝑓(𝑥)𝑓(𝑦)−𝑓(𝑥) = 𝑥𝑦−𝑦U¤ê    ñ
AŽ𝑓(0) = 0
BŽ𝑓(−1) = 1
CŽ𝑓(𝑥+ 1)Ï1
DŽ𝑓(𝑥+ 1)Ï0
°8 4-1ê2024·zğ·¥ßñõ
𝑓(𝑥) =
2−𝑥
2+𝑥U¤ö÷
ÁÏ0
"wê    ñ
AŽ𝑓(𝑥−2) +1
BŽ𝑓(𝑥−2) +2
CŽ𝑓(𝑥+ 2) +2
DŽ𝑓(𝑥+ 2) +1
°8 4-2ê2024·Ćć·ßîïðñ56c~(−∞,0) ∪(0, + ∞)¾"
𝑓(𝑥)ãä𝑓(𝑥𝑦) =
𝑓(−𝑥)
𝑦
+
𝑓(−𝑦)
𝑥
+
1
𝑥𝑦U¤ê    ñ
AŽ𝑓(𝑥)w0
ç~(0, + ∞)¾ ò¡
BŽ𝑓(𝑥)w0
ç~(−∞,0)¾ òŸ
CŽ𝑓(𝑥)w1
ç~(0, + ∞)¾ ò¡
DŽ𝑓(𝑥)w1
ç~(−∞,0)¾ òŸ
°8 4-3ê2024·ĆìĠ·µßñ56
𝑓(𝑥)ãä𝑓(𝑥+ 𝑦+ 1) = 𝑓(𝑥) +𝑓(𝑦)U¤ö÷dáx´¨"
wê    ñ
AŽ𝑓(𝑥) +1w0
BŽ𝑓(𝑥−1)w0
CŽ𝑓(𝑥)−1w0
DŽ𝑓(𝑥+ 1)w0
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é 5ê2024·ĊúúĖ·ßîïðñ56
𝑓(𝑥) = 𝑎−
1
2𝑥−1(𝑎∈𝐑)Ï0
U¤`
𝑎"/Ïê    ñ
AŽ
1
2
BŽ−
1
2
CŽ1
DŽ−1
      
      
°8 5-1ê2024·ġĢģü·¥ßñâ
𝑓(𝑥) =
(𝑥+1)(𝑥+𝑎)
𝑥
Ï0
U¤`
𝑎= ê    ñ
AŽ1
BŽ−1
CŽ2
DŽ−2
°8 5-2ê2024·ĤĥùĦĦħĨ·xßñ56𝑓(𝑥) = 𝑥3 + 2𝑥2,𝑥≥0
𝑥3 + 𝑎𝑥2,𝑥< 0
 
Ï0
U¤𝑎= ê    ñ
AŽ−2
BŽ2
CŽ1
DŽ−1
°8 5-3ê2024·ĩĪ·xßñ56
𝑦= 𝑓(𝑥)wc~𝑅¾"0
Uçý𝑥< 0…U𝑓(𝑥) = 𝑥2 +
𝑎
𝑥Uâ
𝑓(3) = −8U¤𝑎= ê    ñ
AŽ−3
BŽ3
CŽ
1
3
DŽ−
1
3
 6  56
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é 6ê2024·ęúīĬ·xßñ56
𝑓(𝑥)ϐc~ R ¾"0
Uçý𝑥≥0…U𝑓(𝑥) = 2𝑥+𝑥−1U¤
ý𝑥< 0…U𝑓(𝑥) = ê    ñ
AŽ2−𝑥−𝑥−1
BŽ2−𝑥+𝑥+ 1
CŽ−2−𝑥−𝑥−1
DŽ−2−𝑥+𝑥+ 1
°8 6-1ê2024·ĝĞ·ßîïðñâ
𝑓(𝑥) = 𝑓(𝑥+ 2),𝑥≥0
ℎ(𝑥),𝑥< 0
 
"TG»£K:;Uç𝑓(5) = 1U¤ℎ
(−2022) +ℎ(−2023) +ℎ(−2024) = ê    ñ
AŽ−1
BŽ0
CŽ1
DŽ2
°8 6-2ê2024·ĭëúĪ·µßñâ𝑓(𝑥)wc~R¾"0
Uç𝑓(𝑥+ 1)w1
Uý0 < 𝑥≤1…U
𝑓(𝑥) = e𝑥−1U¤ý2 < 𝑥≤3…U𝑓(𝑥)")78Ïê    ñ
AŽ𝑓(𝑥) = −e1−𝑥
BŽ𝑓(𝑥) = −e𝑥+1
CŽ𝑓(𝑥) = −e𝑥−3
DŽ𝑓(𝑥) = −e3−𝑥
°8 6-3ê2024·ëì·¥ßñ56
𝑓(𝑥)Ï0
U𝑔(𝑥)Ï1
Uç𝑓(𝑥)−𝑔(𝑥) = 𝑒𝑥U¤
𝑓(1)
𝑔(1) =
ê    ñ
AŽe2+1
e
BŽe2−1
e
CŽ1−e2
1+e2
DŽ1+e2
1−e2
 7  
":;!2<=!
é 7ê2024·Įìįİ·µßñ56c~𝑅¾"
𝑓(𝑥)w0
U:þb𝑥∈𝑅đÆ𝑓(𝑥+ 1) = 𝑓(1−𝑥)U
ý𝑓(−3) = −2…U¤𝑓(2023)E»ê   ñ
AŽ2
BŽ−2
CŽ0
DŽ−4
°8 7-1ê2024·ıüIJv·¥ßñ56
𝑓(𝑥)"TG~ x 澒U:∀𝑥∈RUđÆ
      
      
𝑓(𝑥+ 2) ⋅𝑓(𝑥) = 2𝑓(1)Uâ𝑦= 𝑓(𝑥−1)"TG»Ûå𝑥= 1:;Uç𝑓(0) = 1U¤
𝑓(2023) + 𝑓(2024) + 𝑓(2025) = ê    ñ
AŽ3
BŽ4
CŽ5
DŽ6
°8 7-2ê2024·|ijĴĵĶķĨ·µßñ56c~ R ¾"
𝑓(𝑥)ãä𝑓(2 + 𝑥)−𝑓(2−𝑥) = 4𝑥Žâ𝑓
(2𝑥−3)"TG»K(2,1):;Uç𝑓(0) = 0U¤𝑓(1) +𝑓(2) + ⋅⋅⋅+𝑓(50) = ê    ñ
AŽ0
BŽ50
CŽ2509
DŽ2499
°8 7-3ê2024·ĸĹ솷¥ßñ56
𝑓(𝑥)、𝑔(𝑥)"c‘ÿÏ RU
𝑓(2𝑥−1) + 1"TG»£
K:;U
𝑔(𝑥+ 1)"TG» y æ:;U𝑓(𝑥+ 2) + 𝑔(𝑥+ 1) = −1,𝑓(−4) = 0U¤𝑓(2030)−𝑔(2017) =
ê    ñ
AŽ−4
BŽ−3
CŽ3
DŽ4
 8  ><=

é 8ê23-24 x¾·ùúĺĖ·=Ļñõ
𝑓(𝑥)"c‘Ï RUç𝑓(𝑥+ 4) = 2𝑓(𝑥)Uý𝑥∈(0,4]…U𝑓(𝑥)
= 2𝑥2−8𝑥Uâ:»∀𝑥∈(−∞,𝑡]UđÆ𝑓(𝑥) ≥−
3
2ØÒÓU¤ t "Ô/Ăăwê    ñ
AŽ(−∞,−7]
BŽ(−∞,−5]
CŽ(−∞,−3]
DŽ(−∞,−1]
°8 8-1ê23-24 x¾·ļùďü·=Áñõ
𝑓(𝑥)"c‘ÏRUãä𝑓(𝑥) = 2𝑓(𝑥−2)Uçý𝑥∈(0,2]
…U𝑓(𝑥) = 𝑥(2−𝑥)Žâ:þb𝑥∈(−∞,𝑚]UđÆ𝑓(𝑥) ≤3U¤𝑚"Ô/Ăăwê    ñ
AŽ −∞, 5
2
BŽ −∞, 7
2
CŽ −∞, 9
2
DŽ −∞, 11
2
°8 8-2ê2024·Ľìľ4·µßñc“
𝑦= 𝑓(𝑥)w𝐷¾"𝑎Ŀ><=
” ŀö: 
𝑦= 𝑓(𝑥),𝑥∈
DU:»½"Ł€Ý
 𝑎UłĐ~Ł€Ý
𝑇U±Öc‘𝐷|"þb`
𝑥đÆ𝑎𝑓(𝑥) = 𝑓(𝑥+ 𝑇)ØÒÓU
х𝑇Ï𝑓(𝑥)"<=. â𝑦= 𝑓(𝑥)w[1, + ∞)¾"𝑎Ŀ><=
Uç𝑇= 1Uý𝑥∈[1,2)…U𝑓(𝑥) = 2𝑥+ 1,ç
𝑦= 𝑓(𝑥)w[1, + ∞)¾" òŸ
U¤`
𝑎"Ô/ĂăÏê    ñ
AŽ 5
6 , + ∞
BŽ[2, + ∞)
CŽ 5
3 , + ∞
DŽ[10, + ∞)
°8 8-3ê23-24 x¾·ńŅńü·=Ļñc‘Ï𝑅"
𝑓(𝑥)ãä𝑓(𝑥+ 1) = 2𝑓(𝑥)Uý𝑥∈[0,2)…U 
𝑓(𝑥) = {
𝑥2−𝑥,𝑥∈[0,1]
log 2(𝑥+ 1),𝑥∈(1,2)Uâ𝑥∈[−2,0)…U:þb"𝑡∈[1,2]đÆ𝑓(𝑥) ≥
𝑡
16 −
𝑎
8𝑡ÒÓU¤`
𝑎"Ô/
Ăăwê    ñ
AŽ(−∞,3)
BŽ[3, + ∞)
CŽ(−∞,3]
DŽ(3, + ∞)
      
      
 9  -
"!?@ABC
é 9ê23-24 x¾·Ćììā·ņŇňʼnñ56c~𝑅¾"
𝑓(𝑥)ãä𝑓(1 + 𝑥) = 𝑓(1−𝑥)Uç∀𝑥1,𝑥2
> 1U𝑥1 ≠𝑥2…U[𝑓(𝑥1)−𝑓(𝑥2)](𝑥1−𝑥2) < 0Uě𝑎= 𝑓
2
2 U𝑏= 𝑓
3
2 U𝑐= 𝑓
6
2 U¤ê    ñ
AŽ𝑏> 𝑐> 𝑎
BŽ𝑏> 𝑎> 𝑐
CŽ𝑐> 𝑏> 𝑎
DŽ𝑐> 𝑎> 𝑏
°8 9-1ê23-24 x¾·ĊúúĖ·=Áñ56
𝑓(𝑥)w1
Uý0 ≤𝑥1 < 𝑥2…U[𝑓(𝑥2)−𝑓(𝑥1)](𝑥2−𝑥1)
> 0ØÒÓUõ𝑎= 𝑓
5 5 U𝑏= 𝑓(−2)U𝑐= 𝑓(
3 3)U¤ aUbUc "BCÍÏê    ñ
AŽ𝑎< 𝑏< 𝑐
BŽ𝑐< 𝑏< 𝑎
CŽb<c<a
DŽ𝑏< 𝑎< 𝑐
°8 9-2ê22-23 x¾·ĸĹÒđ·=Áñ56
𝑦= 𝑓(𝑥)~(0,4)¾ òŸUç𝑦= 𝑓(𝑥+ 4)w1

U¤ê    ñ
AŽ𝑓(2) < 𝑓(3) < 𝑓(7)
BŽ𝑓(7) < 𝑓(2) < 𝑓(3)
CŽ𝑓(2) < 𝑓(7) < 𝑓(3)
DŽ𝑓(3) < 𝑓(7) < 𝑓(2)
°8 9-3ê2023·ĤĥùBğ·ßîïðñ56𝑎= ln1.1U𝑏=
1
11U𝑐=
1.2−1U¤ê    ñ
AŽ𝑎> 𝑏> 𝑐
BŽ𝑎> 𝑐> 𝑏
CŽ𝑐> 𝑏> 𝑎
DŽ𝑐> 𝑎> 𝑏
 10  -
"!?)DE8
é 10ê2024·ĊúúĖ·ßîïðñ56
𝑓(𝑥)"c‘Ï RU:þb`
 xUy đÆ
𝑓(𝑥+ 𝑦) = 𝑓(𝑥) + 𝑓(𝑦)−1Uý𝑥> 0…U𝑓(𝑥) > 1Uç𝑓(2) = 5U¤» x "DE8𝑓(𝑥) + 𝑓(4−3𝑥) < 6")
ŊÏê    ñ
AŽ(1, + ∞)
BŽ(2, + ∞)
CŽ −∞，1
DŽ −∞，2
°8 10-1ê2023·ĆìČā·xßñ56
𝑓(𝑥) = −3𝑥+ 3,𝑥< 0
−𝑥2 + 3,𝑥≥0
 
U¤DE8𝑓(𝑎) > 𝑓(3𝑎−4)")ŊÏ
ê    ñ
AŽ −1
2 , + ∞
BŽ(2, + ∞)
CŽ(−∞,2)
DŽ −∞,−1
2
°8 10-2ê2023·ĊúŋŌ·ßîïðñâ
𝑓(𝑥)wc~𝑅¾"1
U~(−∞,0]¾w¡
Uç𝑓(3)
= 0U¤±Ö𝑓(𝑥) < 0"𝑥"Ô/Ăăwê    ñ
AŽ(−∞,−3)
BŽ(3, + ∞)
CŽ(−3,3)
DŽ(−∞,−3) ∪(3, + ∞)
°8 10-3ê2023·Ćì·ßîïðñâc~(−∞,0) ∪(0, + ∞)¾"
𝑓(𝑥)ž…ãä
①𝑓(𝑥)Ï0
”
②:þb"𝑥1,𝑥2 ∈(0, + ∞)Uç𝑥1 ≠𝑥2UđÆ
𝑥2𝑓(𝑥1)−𝑥1𝑓(𝑥2)
𝑥1−𝑥2
< 0U¤;
𝑓(𝑥)eÆ!?𝑃Ž56
𝑓(𝑥)
      
      
eÆ!?𝑃U¤DE8𝑓(𝑥−2) < 𝑓(𝑥2−4)
𝑥+2 ")ŊÏê    ñ
AŽ(−∞,−1)
BŽ(−3,2)
CŽ(−∞,−3) ∪(−1,2)
DŽ(−∞,−3) ∪(2, + ∞)
 11  FG
" !901!9<=!
é 11ê2024·ęú·xßñ56
𝑓(𝑥)"c‘Ï𝑅U/‘Ï(0, + ∞)Uç:þb𝑚U𝑛∈𝑅UđÆ𝑓(𝑚+ 𝑛)
= 𝑓(𝑚)𝑓(𝑛)Ž𝜑(𝑥) =
𝑓(𝑥)−1
𝑓(𝑥)+1Ž
(1)(𝑓(0)"/Uō34𝜑(𝑥)Ï0
Ž
(2)â𝑥> 0U𝑓(𝑥) > 1Uç𝑓(3) = 4U34𝑓(𝑥)Ï𝑅¾"Ÿ
Uō)DE8𝜑(𝑥) >
15
17Ž
°8 11-1ê23-24 x¾·ćč·=Áñõ
𝑓(𝑥)"c‘w(0, + ∞)Uç:þb´`
 xUy đÆ𝑓(𝑥𝑦)
= 𝑓(𝑥) +𝑓(𝑦)ØÒÓU56𝑓(2) = 1Uçý𝑥> 1…U𝑓(𝑥) > 0.
(1)(𝑓1
2 "/”
(2)#$𝑦= 𝑓(𝑥)~&'(0, + ∞)|" !Uō½«34”
(3))DE8𝑓(2𝑥) > 𝑓(8𝑥−6)−1.
°8 11-2ê2024·ùú·ßîïðñ56
𝑓(𝑥)ãä:xà𝑥1,𝑥2 ∈𝑅đÆ𝑓(𝑥1 + 𝑥2) = 𝑓(𝑥1) +𝑓(𝑥2)−2U
ç𝑓(1) = 0Uý𝑥> 1…Æ𝑓(𝑥) < 0Ž
ê1ñ(𝑓(−1)"/”
ê2ñ#$ō34
𝑓(𝑥)~ R ¾" !”
ê3ñ)DE8
 𝑓(𝑥2−2𝑥)
2 +2𝑓(𝑥2−2𝑥−1)−12 < 0Ž
      
      
°8 11-3ê2024·ùú·ßîïðñ56
𝑝(𝑥)U𝑞(𝑥)"c‘ÿÏ𝑅Uçãä
①∀𝑥> 0U𝑝(𝑥) > 0”
②𝑞(𝑥)Ï1
U𝑞(𝑥) ≥𝑞(0) = 1”③∀𝑥U𝑦∈𝑅U𝑝(𝑥+ 𝑦) = 𝑝(𝑥)𝑞(𝑦) + 𝑞(𝑥)𝑝(𝑦).
(1)(𝑝(0)"/Uō34
𝑝(𝑥)Ï0
”
(2)∀𝑥1,𝑥2 ∈𝑅Uç𝑥1 < 𝑥2U34
①𝑝(𝑥1) = 𝑝𝑥1+𝑥2
2
𝑞𝑥1−𝑥2
2
+𝑞𝑥1+𝑥2
2
𝑝𝑥1−𝑥2
2
”
②𝑝(𝑥) òŸ.
 12  
!?"HIJ
é 12ê2023·ĤĥùŎŏŐ·ßîïðñ56𝑓(𝑥) = 𝑎𝑥2+𝑏𝑥+𝑐
4+𝑥2
wc~[Ĕ2U2]¾"
Uâãä𝑓(𝑥) +𝑓
(−𝑥) = 0ç𝑓(1) =
1
5Ž
(1)(𝑓(𝑥)")78”
(2)õ
𝑔(𝑥) = 𝑥2−2𝑚𝑥+ 4(𝑚∈R)Uâ:þb𝑥1,𝑥2 ∈[1,2]UđÆ𝑔(𝑥2) < 𝑓(𝑥1)ØÒÓU( m "Ô/ĂăŽ
°8 12-1ê23-24 µö·ĤĥùőŒ·=Ļñ56
𝑓(𝑥) =
𝑎𝑥+𝑏
1+𝑥2wc~(−1,1)¾"
U𝑓(−𝑥) = −𝑓
(𝑥)ØÒÓUç𝑓1
2 =
2
5.
(1)¨
𝑓(𝑥)")78”
(2)c34𝑓(𝑥)~(−1,1)¾wŸ
”
(3))DE8𝑓(𝑥−1) +𝑓(𝑥) < 0Ž
      
      
°8 12-2ê2024·ĸĹœĪ·ßîïðñ56
𝑓(𝑥) =
𝑎𝑥+𝑏
𝑥2+1wc~(−1,1)¾"0
Uç𝑓1
2 =
2
5Ž
(1)(
𝑓(𝑥)")78”
(2)#$
𝑓(𝑥)~(−1,1)¾" !Ž
(3))» t "DE8
𝑓𝑡+ 1
2 +𝑓𝑡−1
2 ≤0Ž
°8 12-3ê2024·¾ëŔŕ·xßñ56`
𝑎,𝑏wÝ
U
𝑓(𝑥) = ( 1 + 𝑥+
1−𝑥+𝑎)( 1−𝑥2 +𝑏).
ê1ñ(
𝑓(𝑥)"c‘U#$
"01!UōŖ4ª”
ê2ñâ𝑎= −3,𝑏= 1Uõ𝑡=
1 + 𝑥+
1−𝑥Uě𝑡"Ô/ŗÒ"ŊIÏ𝐷U¤
𝑓(𝑥)"/‘2
𝑔(𝑡) =
1
2
(𝑡3−3𝑡2)(𝑡∈𝐷)"/‘ž.Ř)řö÷Ü
êiñ(ŊI𝐷”
êiiñŚś
𝑔(𝑡) =
1
2(𝑡3−3𝑡2)~c‘𝐷¾wËeÆ !ŜâÆUŝ
 !cŞÈ34”âş
ÆUŝŖ4ª.ō-Š"Śśdš‚xŢ(«
𝑓(𝑥)".C/.
x9
1Žê2024·ĝĞ·ßîïðñâ
𝑓(𝑥) = 4|𝑥−𝑎| + 3~&'[1, + ∞)¾D U¤ a "Ô/Ăăwê    ñ
AŽ[1, + ∞)
BŽ(1, + ∞)
      
      
CŽ(−∞,1)
DŽ(−∞,1]
2Žê2023·ĝĞ·¥ßñ56
𝑓(𝑥) = 𝑏𝑥−(𝑏+ 3)𝑥3~[−1,1]¾".C/Ï−3U¤`
𝑏"Ô/Ăăw
ê    ñ
AŽ(−∞,−4]
BŽ[9, + ∞)
CŽ[−4,9]
DŽ −9
2 ,9
3Žê2024·ĮćţŤ·µßñ56
𝑓(𝑥) = 𝑥|𝑥|U¤»𝑥"DE8𝑓(2𝑥) > 𝑓(1−𝑥)")ŊÏê    ñ
AŽ 1
3 , + ∞
BŽ −∞, 1
3
CŽ 1
3 ,1
DŽ −1, 1
3
4Žê2023·ĊúúĖ·xßñ56𝑓(𝑥)w𝑅¾"0
Uç𝑓(𝑥+ 2) = −𝑓(𝑥)Uý𝑥∈[0,1]…U𝑓(𝑥) = 𝑥2 +2𝑥U
¤𝑓(2023) = ê   ñ
AŽ3
BŽ−3
CŽ255
DŽ−255
5Žê2023·óôťŦ·ßîïðñ56
𝑓(𝑥)"c‘ÏRUâ:∀𝑥∈RđÆ𝑓(3 + 𝑥) = 𝑓(1−𝑥)Uç𝑓(𝑥)~
2，+ ∞¾ ò¡U¤𝑓(1)，𝑓(2)2𝑓(4)"BCÍwê    
ñ
AŽ𝑓(4) < 𝑓(1) < 𝑓(2)
BŽ𝑓(2) < 𝑓(1) < 𝑓(4)
CŽ𝑓(1) < 𝑓(2) < 𝑓(4)
DŽ𝑓(4) < 𝑓(2) < 𝑓(1)
6Žê2023·Ėėŧü·ßîïðñ56
𝑓(𝑥)wc~R¾"1
U
𝑔(𝑥)wc~R¾"0
Uç
𝑓(𝑥)U𝑔(𝑥)~[0, + ∞)¾ ò¡U¤ê    
ñ
AŽ𝑓(𝑓(2)) > 𝑓(𝑓(3))
BŽ𝑓(𝑔(2)) < 𝑓(𝑔(3))
CŽ𝑔(𝑔(2)) > 𝑔(𝑔(3))
DŽ𝑔(𝑓(2)) < 𝑔(𝑓(3))
7Žê2023·óô·xßñ56
𝑓(𝑥)wc~𝑅¾"0
Uý𝑥> 0…U𝑓(𝑥) = 𝑎𝑥+ 1Uâ𝑓(−2) = 5U¤
DE8𝑓(𝑥) >
1
2")ŊÏê    ñ
AŽ −∞,−1
2 ∪0, 1
6
BŽ −1
2 ,0 ∪0, 1
6
CŽ −∞,−1
2 ∪
1
6 , + ∞
DŽ −1
2 ,0 ∪
1
6 , + ∞
8Žê2023·ĝĞ·ßîïðñ56
𝑓(𝑥)U𝑔(𝑥)"c‘ÿÏ𝑅U𝑓(3𝑥+ 1)Ï0
U𝑔(𝑥+ 2)Ï1
U
𝑓(𝑥+ 1) + 𝑔(1−𝑥) = 2U𝑓(0) = −
1
2U¤
102
𝑘=1
𝑔(𝑘) = ê    ñ
AŽ−51
BŽ
5
2
CŽ
415
2
DŽ
409
2
µ9Ũ
9Žê2023·Ćì·ßîïðñ56
𝑓(𝑥)~ R ¾ òŸU
𝑔(𝑥)~(−∞,0)¾ òŸU~[0, + ∞)¾
      
      
 ò¡U¤ê     ñ
AŽ
𝑓(𝑓(𝑥))~ R ¾ òŸ
BŽ
𝑓(𝑔(𝑥))~(−∞,0)¾ òŸ
CŽ
𝑔(−𝑔(𝑥))~(−∞,0)¾ ò¡
DŽ
𝑔(−𝑓(𝑥))~[0, + ∞)¾ ò¡
10Žê2024·ĝĞ·ßîïðñc~(0, + ∞)¾"
𝑓(𝑥)ãäö÷¶·
ê1ñ𝑓𝑥
𝑦= 𝑦𝑓(𝑥)−𝑥𝑓(𝑦)”ê2ñ
ý𝑥> 1…U𝑓(𝑥) > 0U¤ê    ñ
AŽ𝑓(1) = 0
BŽý0 < 𝑥< 1…U𝑓(𝑥) < 0
CŽ𝑓(𝑥2) ≥2𝑓(𝑥)
DŽ𝑓(𝑥)~(1, + ∞)¾ ò¡
11Žê2024·ıüıā·µßñc~𝑅¾"
𝑓(𝑥)ãä𝑓(𝑥+ 1) = 𝑔(𝑥)−𝑔(−𝑥) + 3 = 𝑓(3−𝑥)U:∀𝑥1,𝑥2
∈(1,2]U𝑥1 ≠𝑥2UØÆ
𝑓(𝑥1)−𝑓(𝑥2)
𝑥2−𝑥1
< 0U¤ö÷ũwMũ"Æê    ñ
AŽ(2025,3)w𝑓(𝑥)TG"xy:;Áè BŽ𝑓(𝑥)~&'(2024,2026)¾ ò¡
CŽ:∀𝑥∈
4051
2
, 4053
2
UØÆ𝑓(𝑥) > 𝑓(𝑥+ 1) DŽ
2026
𝑛=1
𝑓(𝑛) > 6078
¥9Ūū
12Žê2024·ĭëúĪ·ßîïðñ56
𝑓(𝑥)wc~𝑅¾"1
Uçãä𝑓(𝑥+ 4) = 𝑓(𝑥)Uý𝑥∈[−2,0]
…U𝑓(𝑥) = −3𝑥−2𝑥U¤𝑓(1) +𝑓(4) =      
.
13Žê2023·¾ëŬŭ·µßñ56
𝑓(𝑥) = 𝑥+
𝑎
𝑥+𝑏U𝑥∈[𝑏, + ∞)U®Á𝑏> 0U𝑎∈RUâ𝑓(𝑥)".C/
Ï 2U¤`
𝑎"Ô/Ăăw      
.
14Žê2024·Ććِ·µßñ56
𝑓(𝑥)"c‘𝐷= (−∞,0) ∪(0, + ∞)U:þb𝑥1,𝑥2 ∈𝐷UØÆ𝑓(𝑥1𝑥2)
−𝑥1−𝑥2 = 𝑥1𝑓(𝑥2) + 𝑥2𝑓(𝑥1)−1Uçý𝑥1 > 𝑥2 > 0…U
𝑥1𝑓(𝑥2)−𝑥2𝑓(𝑥1)
𝑥1−𝑥2
> −1ØÒÓU𝑓(2) = −3U¤DE8
(𝑥+ 1)𝑓
1
𝑥+1 +𝑥+ 2 > 𝑓(−1)")ŊÏ      
.
ĸ9)ů
15Žê2023·ĮćŔŰ·ßîïðñõ𝑎> 0U𝑏> 0U
𝑓(𝑥) = 𝑎−2𝑏+ 2𝑏𝑥−𝑎𝑥2.
(1)(»𝑥"DE8𝑓(𝑥) > 0)Ŋ”
(2)â𝑓(𝑥)~[0,2]¾".C/Ï𝑎−2𝑏U(
𝑏
𝑎"Ô/Ăă.
      
      
16Žê2023·ĺűįŲ·xßñ
𝑓(𝑥)"c‘Ï(0, + ∞)U:»∀𝑥U𝑦∈(0, + ∞)U𝑓(𝑥𝑦) = 𝑓(𝑥) + 𝑓(𝑦)U
çý𝑥> 1…U𝑓(𝑥) < 0Ž
(1)34
𝑓(𝑥)Ï¡
”
(2)â𝑓1
2 = 2U(DE8𝑓(𝑥) + 𝑓(𝑥−1) + 2 > 0")ŊŽ
17Žê22-23 x¾·ĩĪ·=Áñ56
𝑓(𝑥) =
2𝑎𝑥+𝑏
𝑥2+𝑏𝑥+𝑎wc~(−1,1)¾"0
,ç𝑓1
2 =
4
5Ž
(1)¨
𝑓(𝑥)")78;
(2)ý𝑥∈(−1,1)…,#$
𝑓(𝑥)" !,ō34;
(3))DE8𝑓(2𝑥+ 1) +𝑓1
2 𝑥< 0Ž
18Žê2023·Ćì·ßîïðñ56
𝑓(𝑥):þb`
𝑥,𝑦ØÆ𝑓(𝑥−𝑦) + 𝑓(𝑥+ 𝑦) = 𝑓(2𝑥)ÒÓUçý𝑥< 0…U
𝑓(𝑥) > 0.
(1)(𝑓(0)"/;
(2)#$𝑓(𝑥)" !Uō34;
(3))»𝑥"DE8:𝑓𝑥2−(𝑎+ 2)𝑥+𝑓(𝑎+ 𝑦) + 𝑓(𝑎−𝑦) > 0.
      
      
19Žê2023·¾ëŋę·xßñ56
𝑓(𝑥) = 𝑥2−𝑎𝑥−𝑎U𝑎∈R.
(1)#$
𝑓(𝑥)"01!”
(2)â
𝐹(𝑥) = 𝑥⋅𝑓(𝑥)~𝑥= 1ųÆ¿/Uç» x "’Ù𝐹(𝑥) = 𝑚Æ 3 yDž"`*U(`
 m "Ô/Ă
ă”
(3)ě𝑔(𝑥) = −e𝑥êewŴŵ:
"Ŷ
ñ.â:þb𝑥19𝑥2 ∈[0,e]ç𝑥1 > 𝑥2…UÿÆ|𝑓(𝑥1)−𝑓(𝑥2)| <
|𝑔(𝑥1)−𝑔(𝑥2)|ÒÓU(`
 a "Ô/Ăă.