find the range of the function f, of, x, equals, minus, x, squared, minus, 4, x, minus, 4, .f(x)=−x 2 −4x−4.

The range of a function is the set of all possible output values (y-values) that we can get by plugging in the input values (x-values) into the function.

The function given is a quadratic function, f(x) = -x^2 - 4x - 4.

Quadratic functions have a parabolic graph. The coefficient of the x^2 term is negative, which means the parabola opens downwards.

The maximum value of the function is found at the vertex of the parabola. The x-coordinate of the vertex of a parabola given in the form f(x) = ax^2 + bx + c is given by -b/2a.

Here, a = -1 and b = -4. So, the x-coordinate of the vertex is -(-4)/2*(-1) = 2.

Substitute x = 2 into the function to find the maximum value:

f(2) = -(2)^2 - 4*(2) - 4 = -4 - 8 - 4 = -16.

Since the parabola opens downwards, the maximum value is the y-coordinate of the vertex, and there is no minimum value.

So, the range of the function f(x) = -x^2 - 4x - 4 is (-∞, -16].