which of the following accurately shows the range of the function defined below?\nf(x)=\begin{cases}3, & x <…

which of the following accurately shows the range of the function defined below?\nf(x)=\begin{cases}3, & x < 0\\x^{2}+2, & 0leq x < 2\\\frac{1}{2}x + 5, & xgeq 2end{cases}\n \text{o } 3,infty)\n \text{o } 2,infty)\n \text{o } 3,6\n \text{o } (-infty,infty)

which of the following accurately shows the range of the function defined below?\nf(x)=\begin{cases}3, & x < 0\\x^{2}+2, & 0leq x < 2\\\frac{1}{2}x + 5, & xgeq 2end{cases}\n \text{o } 3,infty)\n \text{o } 2,infty)\n \text{o } 3,6\n \text{o } (-infty,infty)

Answer

Answer:

A. $[3,\infty)$

Explanation:

Step1: Analyze $x < 0$ case

When $x<0$, $f(x)=3$.

Step2: Analyze $0\leq x<2$ case

The function is $y = x^{2}+2$. The minimum value occurs at $x = 0$, $y(0)=0^{2}+2=2$, and as $x$ approaches $2$, $y(2^{-})=2^{2}+2=6$. So the range for this part is $[2,6)$.

Step3: Analyze $x\geq2$ case

The function is $y=\frac{1}{2}x + 5$. When $x = 2$, $y=\frac{1}{2}\times2+5=6$, and as $x$ increases, $y$ also increases. So the range for this part is $[6,\infty)$.

Step4: Combine ranges

Combining all parts, the overall range is $[3,\infty)$ since the value $3$ from the first - part and the ranges from the other parts together form this interval.