Final answer:
The real numbers that are in both the range of f(x) = 3x² - 2 and g(x) = 3 - 2x² are [-2, 3], since the range of f(x) is [-2, ∞) and the range of g(x) is (-∞, 3]. These overlap between -2 and 3.
Step-by-step explanation:
To find which real numbers are in both the range of f(x) = 3x² - 2 and g(x) = 3 - 2x², let's analyze their respective ranges. A function's range is the set of all possible output values it can produce.
For f(x), the function is a parabola opening upwards since the coefficient of x² is positive. The vertex form of a parabola is y = a(x - h)² + k, where (h, k) is the vertex of the parabola. In this case, there is no (x - h) term, so the vertex is at x = 0, and the minimum value of f(x) at this vertex is f(0) = -2. Thus, the range of f(x) is [-2, ∞).
For g(x), this is an upside-down parabola since the coefficient of x² is negative. Following similar reasoning, the maximum value of g(x) at the vertex (x = 0) is g(0) = 3. Thus, the range of g(x) is (-∞, 3].
The common real numbers in both ranges will be between the largest minimum and the smallest maximum of the two ranges. Therefore, the common real numbers in both ranges are [-2, 3].