Questions: The domain of the piecewise function is (-∞, ∞). a. Graph the function. b. Use your graph to determine the function's range. f(x)= 1/4 x^2 if x<3 5x-10 if x ≥ 3 a. Choose the correct graph below. xA. B. C b. What is the range of the entire piecewise function? Select the correct choice below and, if necessary, fill in the answer box(es) to complete A. The range does not have any isolated values. It can be described by (Type your answer in interval notation.) B. The range has at least one isolated value. It can be described as the union of the interval(s) and the set . (Use a comma to

Solution Steps

The first piece of the function is $f(x) = \frac{1}{4}x^2$ for $x < 3$. This is a parabola opening upwards. The vertex is at $(0,0)$. Since we are only considering $x < 3$, the graph goes from negative infinity up to $x=3$. When $x=3$, $f(3) = \frac{1}{4}(3^2) = \frac{9}{4} = 2.25$. Since the function is defined for $x<3$ and not at $x=3$, the interval will be $(-∞, 2.25)$.

The second piece is $f(x) = 5x - 10$ for $x \geq 3$. This is a line with a slope of 5. When $x=3$, $f(3) = 5(3) - 10 = 15 - 10 = 5$. Since the function is defined at $x=3$ here, we include 5 in the interval. As $x$ increases beyond 3, $f(x)$ also increases. So the interval is $[5, ∞)$.

The range is the union of the ranges of both pieces. The first piece has a range of $(-∞, 2.25)$, and the second piece has a range of $[5, ∞)$.

Final Answer: The range of the function is $(-∞, 2.25) \cup [5, ∞)$. The correct graph is C, and the range has an isolated point. The range can be described as the union of $(-∞, 2.25)$ and the set $\{5\}$.