select the correct answer. what does this function show at x = 1? f(c) = { x + 3 x < 1; 3x + 1 x >= 1 } a…

Explanation:

Step1: Calculate left - hand limit

For (x\lt1), (f(x)=x + 3). The left - hand limit as (x\to1^{-}) is (\lim_{x\to1^{-}}f(x)=\lim_{x\to1^{-}}(x + 3)=1+3 = 4).

Step2: Calculate right - hand limit

For (x\geq1), (f(x)=3x + 1). The right - hand limit as (x\to1^{+}) is (\lim_{x\to1^{+}}f(x)=\lim_{x\to1^{+}}(3x + 1)=3\times1+1=4).

Step3: Calculate function value at (x = 1)

When (x = 1), (f(1)=3\times1+1=4) since (x = 1) satisfies (x\geq1).

Step4: Check continuity

Since (\lim_{x\to1^{-}}f(x)=\lim_{x\to1^{+}}f(x)=f(1) = 4), the function is continuous at (x = 1).

Answer:

D. continuity