What is the domain and range of f(x)=-x^2-5?
To find the domain and range of a function, we need to consider the set of all possible input values (domain) and the set of all possible output values (range).
For the function f(x) = -x^2 - 5, let's start with the domain.
The domain is the set of all possible input values, or x-values, for which the function is defined. In this case, there are no restrictions on the values we can plug in for x. So, the domain is all real numbers (-∞, +∞).
Next, we determine the range, which is the set of all possible output values, or y-values, generated by the function.
To find the range for a quadratic function, we consider its graph. The given function f(x) = -x^2 - 5 is a downward-facing parabola.
Since it is a quadratic function with a negative leading coefficient, the graph opens downward.
The highest point on the graph (also known as the vertex) corresponds to the maximum value of the function. In this case, the vertex of the parabola is at (0, -5).
Since the graph opens downward, the y-values decrease as you move away from the vertex. This means that the range of the function is (-∞, -5].
Therefore, the domain is all real numbers (-∞, +∞), and the range is (-∞, -5].