the function f(x)=-(x + 5)(x + 1) is shown. what is the range of the function? all real numbers less than or…

Explanation:

Step1: Expand the function

$f(x)=-(x + 5)(x + 1)=-(x^{2}+x+5x + 5)=-x^{2}-6x - 5$

Step2: Find the vertex of the parabola

For a quadratic function $y = ax^{2}+bx + c$, the x - coordinate of the vertex is $x=-\frac{b}{2a}$. Here $a=-1$, $b=-6$, so $x =-\frac{-6}{2\times(-1)}=- 3$. Substitute $x = - 3$ into the function: $f(-3)=-(-3)^{2}-6\times(-3)-5=-9 + 18-5 = 4$. Since $a=-1<0$, the parabola opens downwards.

Answer:

all real numbers less than or equal to 4