Final answer:
To find the range of the product u(x)v(x), determine the range of u(x) and v(x), and consider their possible combinations. The range of (uv)(x) is (-∞, 0) U (0, ∞) excluding 0.
Step-by-step explanation:
To find the range of the product u(x)v(x), we need to consider the possible values of u(x) and v(x).
First, let's determine the range of u(x). Since u(x) is a quadratic function, its graph is a parabola. The coefficient of the x^2 term is -2, which means the parabola opens downward. Therefore, the maximum value of u(x) occurs at the vertex of the parabola. The x-coordinate of the vertex is -b/2a, where a and b are the coefficients of the quadratic term and the linear term, respectively. In this case, a = -2 and b = 0, so the vertex is at x = 0. Substituting this value into u(x), we get u(0) = -2(0)^2 + 3 = 3. Therefore, the range of u(x) is (-∞, 3].
Next, let's determine the range of v(x). The function v(x) is a rational function with x in the denominator. The denominator cannot be zero, so x ≠ 0. Therefore, v(x) can take any value except 0, which means the range of v(x) is (-∞, 0) U (0, ∞).
Finally, to find the range of u(x)v(x), we consider the possible combinations of values from the ranges of u(x) and v(x). Since u(x) ranges from -∞ to 3 and v(x) ranges from (-∞, 0) U (0, ∞), the product u(x)v(x) can take any value from (-∞, 0) U (0, ∞) except 0. Therefore, the range of (uv)(x) is (-∞, 0) U (0, ∞) excluding 0.