Select the correct answer.Consider function f.𝑓(𝑥)={2𝑥,𝑥<0-𝑥2−4𝑥+1,0<𝑥<212𝑥+3,𝑥>2Which statement is true about function f? A. The function is continuous. B. The function is increasing over its entire domain. C. The domain is all real numbers. D. As x approaches positive infinity, 𝑓(𝑥) approaches positive infinity.
Question
Select the correct answer.Consider function f.𝑓(𝑥)={2𝑥,𝑥<0-𝑥2−4𝑥+1,0<𝑥<212𝑥+3,𝑥>2Which statement is true about function f? A. The function is continuous. B. The function is increasing over its entire domain. C. The domain is all real numbers. D. As x approaches positive infinity, 𝑓(𝑥) approaches positive infinity.
Solution
To answer this question, we need to analyze each statement one by one.
A. The function is continuous. To be continuous, a function must be defined and continuous at every point in its domain. Here, the function is not defined at x=0 and x=2, so it is not continuous.
B. The function is increasing over its entire domain. An increasing function is one where the value of f(x) increases as x increases. Here, the function is increasing for x<0 and x>2, but decreasing for 0<x<2. So, the function is not increasing over its entire domain.
C. The domain is all real numbers. The domain of a function is the set of all possible input values (x-values). Here, the function is not defined at x=0 and x=2, so the domain is not all real numbers.
D. As x approaches positive infinity, 𝑓(𝑥) approaches positive infinity. As x approaches positive infinity, the function 𝑓(𝑥) = 12x + 3 also approaches positive infinity.
So, the correct answer is D. As x approaches positive infinity, 𝑓(𝑥) approaches positive infinity.
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