专题2.7函数与方程八大题型（原卷版）_11.4-2_2026-11-04

      
      
专题2.7 函数与方程【八大题型】

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õ 142024·ö÷·øùúû5̉
𝑓(𝑥) = 3𝑥+𝑥−6•g* 𝑥= 𝑥0J˜𝑥0üꪫý*#$4    5
AΠ1
2 ,1
BΠ1, 3
2
CΠ3
2 ,2
DΠ2, 5
2
Ôm 1-142024·þÿ·øùúû5
𝑓(𝑥) = 2𝑥−1 +𝑥−3% !"%#$\4    5
AŒ(−1,0)
BŒ(0,1)
CŒ(1,2)
DŒ(2,3)
Ôm 1-242024·ĀāĂă·gø5>?log3𝑥+ 𝑥= 2%+!"#$\4    5
AŒ(0,1)
BŒ(1,2)
CŒ(2,3)
DŒ(3,4)
Ôm 1-342024·Ąą·øùúû5ð
𝑓(𝑥) = e𝑥−|𝑥−𝑎|,𝑎∈𝑅J˜4    5
AŒ—𝑓(𝑥)"#$4-2J-15œ4-1J05ƕ J˜"#$40J15r• 
BŒ—𝑓(𝑥)"#$4-2J-15œ4-1J05ƕ J˜"#$40J15ć• 
CŒ—𝑓(𝑥)"#$4-2J-15œ4-1J05Ćć• J˜"#$40J15• 
DŒ—𝑓(𝑥)"#$4-2J-15œ4-1J05Ćć• J˜"#$40J15rć• 
      
      
 2  (
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õ 242024·Ĉĉ·gø5
𝑓(𝑥) = sin 2𝑥+ π
3 "#$(0,2π)_% *
©4    5
AŒ2
BŒ3
CŒ4
DŒ5
Ôm 2-142024·ĊÿċČ·øùúû5
𝑦= (𝑥−2)(2𝑥+ 1)% \4    5
AŒ2
BŒ(2,0)
CŒ-2
DŒ2 )-1
Ôm 2-242024·_čĎ·ďø5̉Đ
𝑓(𝑥)%8đĒ© R,𝑓(𝑥+ 3) = −𝑓(−𝑥)J¹𝑓(2) = 0J˜𝑓(𝑥)
"[0,6]% *
%Û:3©4    5
AŒ7
BŒ9
CŒ10
DŒ12
Ôm 2-342024·ēĔĕĖ·gø58đ"𝑅%Đ
𝑓(𝑥)Þß𝑓(1 + 𝑥) = 𝑓(1−𝑥)J¹Ó𝑥∈[0,1]ėJ𝑓(𝑥)
=
1
2sin
π
2𝑥J˜
𝑔(𝑥) = 𝑓(𝑥)−
1
𝑥−4"[−2,10]!• %œ©4    5
AŒ16
BŒ32
CŒ36
DŒ48
 3  +,
 *
(-

õ 342024·ēĔ_Ĉ·ďø5—
𝑓(𝑥) =
ln𝑥
𝑥−
𝑥
𝑚•š* J˜Ę
𝑚%Ù3./©4    5
AŒ(0,e)
BŒ(e, + ∞)
CŒ(0,2e)
DŒ(2e, + ∞)
Ôm 3-142024·ęĚěĜ·øùúû5̉
𝑓(𝑥) = 1−2sin2 𝜔𝑥+ π
6 (𝜔> 0)" 0, π
2 •¹¶•š*
 J˜𝜔%Ù3./\4    5
AΠ7
6 , 13
6
BΠ7
6 , 13
6
CΠ7
6 , 13
6
DΠ7
6 , 13
6
Ôm 3-242024·ö÷ĝĞ·øùúû5̉
𝑓(𝑥) =
𝑥ln𝑥,𝑥> 0,
−1,𝑥= 0,
𝑥ln(−𝑥)−2,𝑥< 0.
 
—ê𝑥%>?𝑓(𝑥) = 𝑎𝑥−1•
5 *µÂ%Ę
+J˜𝑎%Ù3./\4    5
AŒ(1, + ∞)
BŒ(2, + ∞)
CŒ(1,e)
DŒ(2,2e)
Ôm 3-342024·ęĚğÁ·Qø5̉
𝑓(𝑥) =
1
2
𝑥
ln 1
2 ,𝑥≤0
4ln2𝑥,𝑥> 0
 
J—
𝑔(𝑥) = 𝑓(𝑥)−𝑚𝑥• 4 * J
˜𝑚%Ù3./©4    5
AŒ 𝑚|𝑚≥16
e2
 
BŒ 𝑚|𝑚≥eln22 
CŒ 𝑚|eln22 < 𝑚< 16
e2
 
DŒ 𝑚|𝑚= eln22 或𝑚= 16
e2
      
      
 4  +,
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õ 442024·ēĔŸĆ·ďø5—
𝑓(𝑥) = e𝑥−𝑘𝑥29ê0% •¹È•g*J˜Ę
𝑘%3©      
Œ
Ôm 4-142024·Ąą·øùúû5
𝑓(𝑥) = (𝑥+ 2)ln(𝑥+ 1)−𝑎𝑥ȕ 3 * 𝑥1J𝑥2J𝑥3 
(𝑥1 < 𝑥2 < 𝑥3 < 3)J˜𝑎+ 𝑥2%Ù3./\      
Œ
Ôm 4-242024·ęĚĚě·Qø5̉
𝑓(𝑥) = 3cos(𝜔𝑥+ 𝜑)(𝜔> 0)J—𝑓−π
4 = 3J𝑓π
2 = 0J¹
𝑓(𝑥)"#$ −π
3 ,−π
6 ć• J˜𝜔%g*Ù3©      
Œ
Ôm 4-342024·VW·Qø5ð𝑎∈𝑅J
𝑓(𝑥) =
|
1
𝑥−𝑎+1 + 𝑎|,𝑥< 𝑎
𝑥2−2(𝑎+ 1)𝑥+ 2𝑎2−𝑎+ 1,𝑥≥𝑎
 
. —𝑓(𝑥)"#$
[0, + ∞)_Ġ• 2 * J˜𝑎%Ù3./\      
.
 5  0
 12(34./5
õ 542024·Ąą·øùúû5̉
𝑓(𝑥) = 𝑥+
𝑚
𝑥−2,𝑥∈(0, + ∞)•š* J˜Ę
𝑚%Ù3./\
4    5
AŒ(−∞,0]
BŒ(−∞,1]
CŒ[−1, + ∞)
DŒ(0,1)
Ôm 5-142024·þġĈ·Qø5̉Ģ*ďģc%ďĤĂ©𝑎=𝑏ĥ𝑐J¤Á𝑎< 𝑏.—𝑎J𝑏\
𝑦= 𝑎𝑥2
−𝑏𝑥+ 𝑐%š* J˜𝑎%Ù3./\4    5
AΠ1
2 ,1
BΠ1
2 ,
5−1
2
CΠ0,
5−1
2
DŒ
5−1
2
,1
Ôm 5-242023·Ħÿ·Qø5ð𝑥1,𝑥2\ê x %>?𝑥2 + (𝑎−1)𝑥+ 𝑎+ 2 = 0%+Œ—−1 < 𝑥1 < 1,1 < 𝑥2
< 2J˜Ę
 a %Ù3./\4    5
AŒ −4
3 ,−1
BŒ −3
4 , 1
2
CŒ(−2,1)
DŒ(−2,−1)
Ôm 5-342023·Ąą·øùúû5̉
𝑓(𝑥) =
1
𝑥−𝑥,𝑥> 0
𝑎𝑥2 + 2𝑎𝑥+ 3,𝑥≤0
 
•¹¶• 3 * 𝛼,𝛽,𝛾J—
𝛼< 𝛽< 𝛾J˜4    5
AŒln𝛼𝛽= 𝛾
BŒln𝛼𝛽= 𝛾−1
CŒln𝛼𝛽< 𝛾−1
DŒln𝛼𝛽> 𝛾
 6  67
% *
&8
õ 642024·ħĈĨĩ·ďø5—
𝑓(𝑥) = 𝑥+
1
|𝑥|J˜>?𝑓[𝑓(𝑥)] = 3%Ę
+*
©4    5
AŒ2
BŒ3
CŒ4
DŒ5
Ôm 6-142024·ĪÍīĬ·øùúû5̉
𝑓(𝑥) =
ln𝑥−1
𝑥,𝑥> 0
−|𝑥+ 1| + 1,𝑥≤0
 
J˜
𝑔(𝑥) = 𝑓(𝑓(𝑥)−1)%
      
      
 *
©4    5
AŒ3
BŒ5
CŒ6
DŒ8
Ôm 6-242024·ęĚĭČ·øùúû5̉
𝑓(𝑥) =
−𝑥e𝑥,
𝑥< 0
ln(𝑥+ 1),𝑥≥0
 
J˜
𝑔(𝑥) = 𝑓(𝑓(𝑥))−1%
 *
\4     5
AŒ1
BŒ0
CŒ2
DŒ3
Ôm 6-342024·Ąą·Qø5̉
𝑓(𝑥) =
1
1−𝑥,𝑥< 0
|ln𝑥|,𝑥> 0
 
4e©ĕĮé
%į
5J˜
𝐹(𝑥) = 𝑓[𝑓(𝑥)]
−
1
e3𝑓(𝑥)−1% *
©4    5
AŒ3
BŒ5
CŒ7
DŒ9
 7  +,67
 (-

õ 742024·Ěİāı·øùúû5̉
𝑓(𝑥)%8đĒ©𝑅J𝑓(𝑥) =
−𝑥2,𝑥< 0
𝑥,0 ≤𝑥≤1
2−𝑥,1 < 𝑥≤2
𝑓(𝑥−2),𝑥> 2
 
J—
𝑔(𝑥) = 𝑓
(𝑥)−𝑚𝑥•ď* J˜Ę
𝑚%Ù3./©      
.
Ôm 7-142024·Ąą·øùúû5̉
𝑓(𝑥) =
|4𝑥−1|,𝑥≤1
𝑥2−6𝑥+ 8,𝑥> 1
 
J—>?2[𝑓(𝑥)]2−(𝑎+ 2) ⋅𝑓(𝑥) +𝑎= 0
• 7 *µÂ%Ę
+J˜Ę
𝑎%Ù3./\      
Œ
Ôm 7-242024·VWIJþ·Qø5̉
𝑓(𝑥) =
|𝑥−1|−1,𝑥≤2
−1
2 𝑓(𝑥−2),𝑥> 2
 
J—
𝑔(𝑥) = 𝑥⋅𝑓(𝑥)−𝑎%
 *
© 2J˜ a %./©      
.
Ôm 7-342024·öÿij·Qø5̉
𝑓(𝑥) =
2𝑥,𝑥≤0,
2ln𝑥
𝑥,𝑥> 0,
 
 𝑔(𝑥) = 𝑥2 +2𝑥−4𝜆J𝜆∈𝑅J—ê𝑥%>
?𝑓(𝑔(𝑥)) = 𝜆• 6 *FJ˜𝜆%Ù3./©      
Œ
 8  
 %9:;./<
õ 842024·Ąą·øùúû5̉
𝑓(𝑥) = 𝑎e𝑥−𝑥(𝑎∈R)•š* .
(1)(Ę
 a %Ù3./p
(2)ð𝑓(𝑥)%š* 1Ĵ©𝑥1,𝑥2JĵĶ
𝑥1 + 𝑥2 > 2p
(3)ĵĶ
2
3 +
2
5 +⋯+
2
2𝑛+1 < ln(𝑛+ 1),𝑛∈N∗.
      
      
Ôm 8-142024·Ąą·øùúû5̉
𝑓(𝑥) = (𝑥2−𝑥+ 1)e𝑥−1−
𝑎
6(2𝑥3 + 3𝑥2 + 1)Œ
(1)Ó𝑎= 2ėJ(·¸𝑦= 𝑓(𝑥)" (1,𝑓(1))ķ%ĸ¸>?Œ
(2)ð
𝑔(𝑥) = 𝑓(𝑥)−𝑥2e𝑥−1 +
1
3𝑎𝑥3J—𝑔(𝑥)•š* J(Ę
𝑎%Ù3./Œ
Ôm 8-242024·ö÷Ĺĺ·ďø5̉
𝑓(𝑥) = 𝑥(e𝑥−𝑎𝑥2)J𝑎∈𝑅Œ
(1)(·¸𝑦= 𝑓(𝑥)" (0,𝑓(0))ķ%ĸ¸>?Œ
(2)̉ê𝑥%>?𝑓(𝑥) = 𝑎𝑥2−e𝑥Ġ• 4 *µÂ%Ę
+𝑥1,𝑥2,𝑥3,𝑥4J¤Á𝑥1 > 0J𝑥2 > 0Œ
4i5(𝑎%Ù3./p
4ii5(ĵ
𝑥1 + 𝑥2 > 4Œ
Ôm 8-342024·ħĈ·Qø5̉
𝑓(𝑥) = 𝑥2−𝑎𝑥ln𝑥−1J𝑎∈𝑅Œ
(1)(ĵ
𝑓(𝑥) + 𝑥2𝑓1
𝑥= 0p
(2)—
𝑓(𝑥)•ď*µÂ% 𝑥1J𝑥2J𝑥3(𝑥1 < 𝑥2 < 𝑥3)Œ
4ⅰ5( a %Ù3./p
4ⅱ5(ĵ
𝑥1 + 𝑥3 > 2𝑎−2Œ
g=N
1Œ42024·ĻļĽľ·Qø5
𝑓(𝑥) = 𝑎𝑥−𝑎(𝑎> 0，𝑎≠1)% ©4   5
      
      
AŒ0
BŒ1
CŒ(1,0)
DŒ𝑎
2Œ42024·ęĚěĜ·øùúû5
𝑓(𝑥) = ln𝑥+ 𝑥2−2% !"#$\4    5
AΠ0,
2
2
BŒ
2
2 ,1
CŒ(1, 2)
DŒ( 2,2)
3Œ42024·ĿŀŁł·øùúû5
𝑓(𝑥) = e𝑥(𝑥−1)−𝑥−1%!• ќ©4    5
AŒ0
BŒ-1
CΠ3
DŒ2
4Œ42024·ħĈńŅ·ďø5̉
𝑓(2𝑥+ 1)©ņ
J—
𝑔(𝑥) = 𝑓(𝑥) + 21−𝑥+ 2𝑥−1−5% *
©Đ
*J˜𝑓(1) = 4    5
AŒ1
BŒ2
CŒ3
DŒ0
5Œ42024·ęĚĚě·øùúû5—
𝑓(𝑥) = 𝑥3−3𝑥+ 𝑎"#$(0,2)_•š* J˜Ę
𝑎%Ù3./\
4    5
AŒ(0,2)
BŒ(2, + ∞)
CŒ(0,1)
DŒ(1, + ∞)
6Œ42024·ŇļňĈ·Qø5̉
𝑓(𝑥) = |2𝑥−1|−𝑎J𝑔(𝑥) = 𝑥2−4|𝑥| +2−𝑎J˜4    5
AŒÓ𝑔(𝑥)• 2 * ėJ𝑓(𝑥)ȕ 1 * 
BŒÓ𝑔(𝑥)• 3 * ėJ𝑓(𝑥)• 2 * 
CŒÓ𝑓(𝑥)• 2 * ėJ𝑔(𝑥)• 2 * 
DŒÓ𝑓(𝑥)• 2 * ėJ𝑔(𝑥)• 4 * 
7Œ42024·ēĔʼnČ·øùúû5̉
 𝑓(𝑥) =
e𝑥,𝑥≤0,
ln𝑥,𝑥> 0,
 
 𝑔(𝑥) = 𝑥−3,>?𝑓(𝑔(𝑥)) = −3−𝑔(𝑥)•š*µ
Â%+J1Ĵ\𝑥1,𝑥2,˜ 𝑥1 + 𝑥2 = 4   5
AŒ0
BŒ3
CŒ6
DŒ9
8Œ42024·ěŊ7ŋ·ďø5ð𝑎∈𝑅J
𝑓(𝑥) = 2|𝑥−1|−1,𝑥≥0
−𝑥2 + 𝑎𝑥,𝑥< 0
 
J—
𝑦= 𝑓(𝑓(𝑥))Ġ• 5 * J˜Ę

𝑎%Ù3./©4    5
AŒ(−2,2)
BŒ(0,2)
CŒ[−1,0)
DŒ(−∞,−2)
Q=¾
9Œ42024·Ōō·gø5̉
𝑓(𝑥) = 2cos 𝜔𝑥+ π
6 +2(𝜔> 0)"#$ −π
6 , π
3 N¼ŎŏJ¹"#$[0,π]
•¹¶•g* J˜𝜔%3 i©4    5
AŒ
2
3
BŒ
5
6
CŒ
11
12
DŒ
13
12
10Œ42024·ö÷·øùúû5̉
𝑓(𝑥) = e𝑥+2𝑥−2,𝑔(𝑥) = 2ln𝑥+ 𝑥−2% 1Ĵ©𝑥1,𝑥2J˜4    5
      
      
AŒ2𝑥1 + 𝑥2 = 2
BŒ𝑥1𝑥2 = e𝑥1 + ln𝑥2
CŒ𝑥1 + 𝑥2 >
4
3
DŒ2𝑥1𝑥2 <
e
11Œ42024·ĈĚŐă·øùúû5̉
𝑓(𝑥) = 2−|log1
2𝑥|,0 < 𝑥≤2
−𝑥2 + 8𝑥−11,𝑥> 2,
 
J𝑔(𝑥) = 𝑓(𝑥)−𝑎J˜4    5
AŒ—𝑔(𝑥)• 2 *µÂ% J˜2 < 𝑎< 5
BŒÓ𝑎= 2ėJ𝑔(𝑓(𝑥))• 5 *µÂ% 
CŒ—𝑔(𝑥)• 4 *µÂ% 𝑥1,𝑥2,𝑥3,𝑥4(𝑥1 < 𝑥2 < 𝑥3 < 𝑥4)J˜𝑥1𝑥2𝑥3𝑥4%Ù3./\(12,13)
DŒ—𝑔(𝑥)• 4 *µÂ% 𝑥1,𝑥2,𝑥3,𝑥4(𝑥1 < 𝑥2 < 𝑥3 < 𝑥4)J˜𝑎𝑥1𝑥2 +
𝑥3+𝑥4
𝑎%Ù3./\(6,9)
ď=kl
12Œ42023·ŌōőŒľ·gø5œŔC
𝑓(𝑥) =
6
𝑥−log2𝑥 !"%g*#$      
.
13Œ42024·VW÷ŕ·ďø5—
𝑓(𝑥) = 𝑎|2𝑥−3|−3𝑎−𝑥2(𝑥−3)2•ē* J˜Ę
𝑎%Ù3./©
      
.
14Œ42024·ö÷Ŗŗľ·ďø5̉Đ
𝑓(𝑥)%8đĒ©𝑅J𝑓(𝑥+ 3) = −𝑓(−𝑥)J¹𝑓(2) = 0J˜𝑓(𝑥)"[0,6]
% *
%Û:3©      
.
ē=FŘ
15Œ42024·ēĔřĬ·ďø5̉
𝑓(𝑥) = 𝑎𝑥e𝑥−14𝑎> 05J
(1)Śś
𝑓(𝑥)% *
p
(2)—|𝑓(𝑥)| > 𝑥+ 𝑥ln𝑥ŜŸŝJ(
𝑓(𝑥)% 𝑥0%Ù3./Œ
16Œ42024·Ąą·øùúû5̉
𝑓(𝑥) = (𝑥−1)2e𝑥−𝑎𝑥J¹·¸𝑦= 𝑓(𝑥)" (0,𝑓(𝑥))ķ%ĸ¸>?©
𝑦= −2𝑥+ 𝑏Œ
(1)(Ę
𝑎J𝑏%3p
(2)ĵĶ

𝑓(𝑥)•š* Œ
      
      
17Œ42024·öÿČ·øùúû5ð
𝑓(𝑥) = (𝑥−1)e𝑥−1 +
𝑎
2𝑥2.
(1)Śś
𝑓(𝑥)%N¼yp
(2)—𝑎≥−eJŚś
𝑓(𝑥)% %*
.
18Œ42024·Ċ÷Şş·ďø5̉
𝑓(𝑥) = 𝑥−ln𝑥+ 𝑚•š* 𝑥1J𝑥2.
(1)(Ę
𝑚%Ù3./p
(2)¢°𝑥1 < 𝑥2 ≤2𝑥1J(Šė𝑚%Ù3./.
19Œ42024·Ąą·øùúû5̉
𝑓(𝑥) = 𝑥e
1
𝑥−𝑎(𝑥> 0)J¹𝑓(𝑥)•š*çš 𝑥1,𝑥2Œ
(1)(Ę
 a %Ù3./Œ
(2)ĵĶ
𝑥1 + 𝑥2 >
2𝑎
e Œ