the function f(x)=-x² - 4x + 5 is shown on the graph. which statement about the function is true? the domain…

Explanation:

Step1: Recall domain and range definitions

Domain is all possible x - values, range is all possible y - values for a function. For a quadratic function (y = ax^{2}+bx + c) (here (a=-1), (b = 4), (c = 5)), the domain of any quadratic function (y=ax^{2}+bx + c) is all real numbers since we can substitute any real number for (x).

Step2: Find the vertex of the quadratic function

The x - coordinate of the vertex of a quadratic function (y = ax^{2}+bx + c) is given by (x=-\frac{b}{2a}). Substituting (a=-1) and (b = 4), we have (x =-\frac{4}{2\times(-1)}= - 2). Then substitute (x=-2) into the function (y=-(-2)^{2}-4\times(-2)+5=-4 + 8+5=9). Since (a=-1<0), the parabola opens downwards.

Step3: Determine the range

The maximum value of the function occurs at the vertex. So the range of the function (y=-x^{2}-4x + 5) is all real numbers less than or equal to the y - coordinate of the vertex. The range is (y\leqslant9).

Answer:

The range of the function is all real numbers less than or equal to 9.