Final answer:
The correct answer is option c. The function with a domain of (-∞, ∞) and a range of (-∞,4] is f(x) = -x^2 + 4, which is an inverted parabola with its vertex at (0, 4).
Step-by-step explanation:
The correct answer is option c, the function f(x) = -x2 + 4. This function has a domain of (-∞, ∞) because you can plug any real number into a quadratic function, and a range of (-∞, 4] because it is an inverted parabola with a vertex at the point (0, 4). The vertex represents the maximum value of the function since it opens downwards due to the negative coefficient of the x2 term. This means that the function does not produce any output higher than 4, fulfilling the condition for the range specified in the question.
To determine the function with a domain of (-∞, ∞) and a range of (-∞, 4), we need a function that can take any real number as input (hence the domain of (-∞, ∞)), and outputs a value less than or equal to 4 (hence the range of (-∞, 4)). The function -x^2 + 4 meets these criteria. It is a quadratic function that can take any real number as input and outputs a value less than or equal to 4. The negative coefficient of x^2 ensures that the function is always decreasing, preventing it from going beyond the range of (-∞, 4).