To determine the domain and range of the function [tex]\( f(x) = -x^2 - 2x + 15 \)[/tex], let's analyze it step by step.
### Domain
The domain of a function consists of all the input values (x-values) for which the function is defined. In this case, since [tex]\( f(x) = -x^2 - 2x + 15 \)[/tex] is a quadratic function, it is defined for all real numbers. Therefore, the domain of the function is:
[tex]\[ \text{Domain: } \{ x \mid x \in \mathbb{R} \} \][/tex]
### Range
The range of a function consists of all the output values (y-values) that the function can produce. Since this is a quadratic function with a negative leading coefficient (-1), the parabola opens downwards.
To find the range, we need to identify the maximum value of the function, which occurs at the vertex of the parabola.
The general form of a quadratic function is [tex]\( f(x) = ax^2 + bx + c \)[/tex]. The x-coordinate of the vertex is given by:
[tex]\[ x = -\frac{b}{2a} \][/tex]
For our function [tex]\( f(x) = -x^2 - 2x + 15 \)[/tex], we can identify that:
[tex]\[ a = -1, \quad b = -2, \quad c = 15 \][/tex]
So, the x-coordinate of the vertex is:
[tex]\[ x = -\frac{-2}{2 \cdot -1} = \frac{2}{-2} = -1 \][/tex]
Next, we plug [tex]\( x = -1 \)[/tex] back into the function to find the y-coordinate of the vertex:
[tex]\[ f(-1) = -(-1)^2 - 2(-1) + 15 \][/tex]
[tex]\[ f(-1) = -1 + 2 + 15 \][/tex]
[tex]\[ f(-1) = 16 \][/tex]
Thus, the maximum value of [tex]\( f(x) \)[/tex] is 16, and since the parabola opens downward, the range includes all y-values up to and including 16. Therefore, the range is:
[tex]\[ \text{Range: } \{ y \mid y \leq 16 \} \][/tex]
### Conclusion
Putting it all together, we get:
- The domain of the function is all real numbers.
- The range of the function is [tex]\( \{ y \mid y \leq 16 \} \)[/tex].
Thus, the correct statement is:
The domain is all real numbers. The range is [tex]\(\{ y \mid y \leq 16 \} \)[/tex].
