Question

on core algebra i b - cr
the graph of the function ( f(x)=(x + 2)(x + 6) ) is shown below.
what is true about the domain and range of the function?
( \bigcirc ) the domain is all real numbers, and the range is all real numbers greater than or equal to ( - 4 ).
( \bigcirc ) the domain is all real numbers greater than or equal to ( - 4 ), and the range is all real numbers.
( \bigcirc ) the domain is all real numbers such that ( - 6leq xleq - 2 ), and the range is all real numbers greater than or equal to ( - 4 ).
( \bigcirc ) the domain is all real numbers greater than or equal to ( - 4 ), and the range is all real numbers such that ( - 6leq xleq - 2 ).

Explanation:

Step1: Identify the function type

The function $f(x)=(x+2)(x+6)$ is a quadratic function (degree 2 polynomial).

Step2: Find the domain of quadratic functions

For all polynomial functions, the domain is all real numbers, since there are no restrictions on $x$ values that make the function undefined.

Step3: Rewrite function in vertex form

First expand the function:
$f(x)=x^2+8x+12$
Complete the square:
$f(x)=(x^2+8x+16)-16+12=(x+4)^2-4$

Step4: Determine the range

The vertex form $f(x)=(x+4)^2-4$ has a vertex at $(-4,-4)$. Since the coefficient of $(x+4)^2$ is positive, the parabola opens upward. This means the minimum value of the function is $-4$, so the range is all real numbers greater than or equal to $-4$.

Answer:

The domain is all real numbers, and the range is all real numbers greater than or equal to -4.