Let's analyze the function [tex]\( f(x) = -x^2 - 2x + 15 \)[/tex] to determine its domain and range.
### Step 1: Determine the Domain
The function [tex]\( f(x) \)[/tex] is a quadratic function in the form [tex]\( ax^2 + bx + c \)[/tex] where [tex]\( a = -1 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = 15 \)[/tex]. For quadratic functions, the domain is all real numbers because there are no restrictions on the values [tex]\( x \)[/tex] can take. Thus,
Domain: All real numbers.
### Step 2: Determine the Range
To find the range, we need to locate the vertex of the parabola described by the quadratic function. The vertex of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] can be found using the formula for the x-coordinate of the vertex:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Given [tex]\( a = -1 \)[/tex] and [tex]\( b = -2 \)[/tex]:
[tex]\[ x = -\frac{-2}{2(-1)} = -\frac{2}{-2} = 1 \][/tex]
Now, we substitute [tex]\( x = 1 \)[/tex] back into the function [tex]\( f(x) \)[/tex] to find the y-coordinate of the vertex:
[tex]\[ f(1) = -1(1)^2 - 2(1) + 15 = -1 - 2 + 15 = 16 \][/tex]
Since the parabola opens downwards (because [tex]\( a = -1 < 0 \)[/tex]), the vertex represents the maximum value of the function. This value occurs at the y-coordinate 16. Therefore, the range of the function consists of all [tex]\( y \)[/tex]-values less than or equal to this maximum value [tex]\( 16 \)[/tex]:
Range: [tex]\( y \le 16 \)[/tex]
### Conclusion
By combining the domain and range we have determined, we get:
- Domain: All real numbers.
- Range: [tex]\( y \leq 16 \)[/tex]
Thus, the correct answer is:
The domain is all real numbers. The range is [tex]\( \{ y \mid y \leq 16 \} \)[/tex].
