Final answer:
The range of the function h(x)=x^2+2x+4 for the given domain x ≤ -4 and -4 ≤ x ≤ 4 is 12 to 3, inclusive.
Step-by-step explanation:
The function hf is defined as h(x)=x^2+2x+4. To find the range of h corresponding to the given domain, we need to substitute the values of x less than or equal to -4 and x less than or equal to 4 into the function and determine the resulting output values.
For x ≤ -4, h(x) = (-4)^2 + 2(-4) + 4 = 16 - 8 + 4 = 12.
For -4 ≤ x ≤ 4, we can find the minimum and maximum values of the function by evaluating the function at the critical points. The vertex of the quadratic function is given by (-b/2a, f(-b/2a)), where a, b, and c are the coefficients in the equation. In this case, the vertex is (-1, 3).
Therefore, the range of h corresponding to the domain x ≤ -4 and -4 ≤ x ≤ 4 is 12 to 3, inclusive.