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로그인 회원가입 You are solving a Korean CSAT mathematics problem. Reason carefully, but the final response must contain exactly one final answer inside \boxed{}.

For multiple-choice problems, put only one of ①, ②, ③, ④, ⑤ inside \boxed{}. For short-answer problems, put only the final integer inside \boxed{}. Do not use external tools.

Problem 4 Consider the function f(x) = ( 3x−2 (x <1), x2 −3x+a(x≥1), which is continuous on the entire real line. Determine the value of the constanta. [3 points] ①1 ②2 ③3 ④4 ⑤5 더보기

The function 𝑓 ( 𝑥 ) f(x) is continuous at 𝑥

1 x=1 if the left-hand limit, right-hand limit, and function value at 𝑥

1 x=1 are all equal.

For 𝑥 < 1 x<1:

lim ⁡ 𝑥 → 1 − 𝑓 ( 𝑥 )

lim ⁡ 𝑥 → 1 − ( 3 𝑥 − 2 )

3 ( 1 ) − 2

x→1 − lim ​

f(x)= x→1 − lim ​

(3x−2)=3(1)−2=1.

For 𝑥 ≥ 1 x≥1:

𝑓 ( 1 )

1 2 − 3 ( 1 ) + 𝑎

1 − 3 + 𝑎

𝑎 − 2. f(1)=1 2 −3(1)+a=1−3+a=a−2. lim ⁡ 𝑥 → 1 + 𝑓 ( 𝑥 )

𝑎 − 2. x→1 + lim ​

f(x)=a−2.

Set them equal:

1

𝑎 − 2 ⟹ 𝑎

1=a−2⟹a=3.

\boxed{3}

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