Find the range of left-parenthesis f plus g right-parenthesis left-parenthesis x right-parenthesis when lower f left-parenthesis x right-parenthesis equals left-parenthesis x plus 4 right-parenthesis squared and lower g left-parenthesis x right-parenthesis equals 3.

Question

(1 point)
Responses

left-parenthesis negative infinity comma infinity right-parenthesis
Image with alt text: left-parenthesis negative infinity comma infinity right-parenthesis

left-bracket 3 comma infinity right-parenthesis
Image with alt text: left-bracket 3 comma infinity right-parenthesis

left-parenthesis negative infinity comma negative 3 right-parenthesis
Image with alt text: left-parenthesis negative infinity comma negative 3 right-parenthesis

left-bracket negative 3 comma infinity right-parenthesis

Answer 1

To find the range of \((f + g)(x)\) when given \(f(x) = (x + 4)^2\) and \(g(x) = 3\), we first need to determine what \((f + g)(x)\) looks like:

\[ (f + g)(x) = f(x) + g(x) = (x + 4)^2 + 3 \]

Next, we need to analyze the expression \((x + 4)^2 + 3\).

1. The term \((x + 4)^2\) represents a parabola that opens upwards. The minimum value of this term occurs when \(x + 4 = 0\) (i.e., when \(x = -4\)), and in this case, \((x + 4)^2 = 0\).

2. Therefore, the minimum value of \((x + 4)^2\) is \(0\), and when we add \(3\) to this minimum, we find:

\[ (f + g)(-4) = (0) + 3 = 3 \]

3. As \(x\) moves away from \(-4\) in either direction, \((x + 4)^2\) increases, and therefore \((f + g)(x)\) will also increase. This means that \((f + g)(x)\) approaches infinity as \(x\) moves towards positive or negative infinity.

Thus, the range of \((f + g)(x)\) is:

\[ [3, \infty) \]

This leads us to the conclusion that the final answer is:

Left-bracket 3, infinity right-parenthesis.