22 select the correct answer. consider function f. f(x) = { 2^x, x < 0 -x^2 - 4x + 1, 0 < x < 2 1/2x + 3, x…

Answer:

C. The domain is all real numbers.

Explanation:

Step1: Analyze domain of each part

For (y = 2^{x}), (x<0); for (y=-x^{2}-4x + 1), (0 < x<2); for (y=\frac{1}{2}x + 3), (x>2). The union of (x<0), (0 < x<2) and (x>2) along with the non - defined points (x = 0) and (x=2) gives all real numbers as the domain.

Step2: Analyze option A

As (x\rightarrow+\infty), (f(x)=\frac{1}{2}x + 3), which approaches (+\infty), but we need to consider the whole function behavior. For (0 < x<2), (y=-x^{2}-4x + 1) is a downward - opening parabola, so the statement "As (x) approaches positive infinity, (f(x)) approaches positive infinity" is not a comprehensive description of the whole function.

Step3: Analyze option B

Check continuity at (x = 0) and (x=2). At (x = 0), (\lim_{x\rightarrow0^{-}}2^{x}=1), (\lim_{x\rightarrow0^{+}}-x^{2}-4x + 1 = 1). At (x = 2), (\lim_{x\rightarrow2^{-}}-x^{2}-4x + 1=-4 - 8+1=-11), (\lim_{x\rightarrow2^{+}}\frac{1}{2}x + 3=1 + 3=4). The function is not continuous.

Step4: Analyze option D

The function (y=-x^{2}-4x + 1) for (0 < x<2) is a downward - opening parabola ((y=-(x + 2)^{2}+5)), so it is decreasing on part of its sub - domain, and the function is not increasing over its entire domain.