Let's evaluate the domain and range of the given piecewise function:
[tex]\[ f(x) =
\begin{cases}
-x - 2 & \text{for } x < -2 \\
-x^2 & \text{for } -2 < x < 0 \\
x & \text{for } x \geq 0
\end{cases}
\][/tex]
### Domain of [tex]\( f(x) \)[/tex]
The domain of a function is the set of all permissible inputs (x-values). In this function, there are no restrictions such as division by zero or taking the square root of a negative number that would restrict the domain. Thus, the domain includes all real numbers, but we need to consider the intervals defined by the piecewise function:
- For [tex]\( x < -2 \)[/tex]: The function [tex]\( f(x) = -x - 2 \)[/tex] is defined.
- For [tex]\( -2 < x < 0 \)[/tex]: The function [tex]\( f(x) = -x^2 \)[/tex] is defined.
- For [tex]\( x \geq 0 \)[/tex]: The function [tex]\( f(x) = x \)[/tex] is defined.
This covers all real numbers except at [tex]\( x = -2 \)[/tex], where the function is not explicitly defined.
Hence, the domain is:
[tex]\[ (-\infty, -2) \cup (-2, \infty) \][/tex]
### Range of [tex]\( f(x) \)[/tex]
The range of a function is the set of all possible output values (y-values).
- For [tex]\( x < -2 \)[/tex]: The function is [tex]\( f(x) = -x - 2 \)[/tex]. As [tex]\( x \)[/tex] approaches [tex]\(-\infty\)[/tex], [tex]\( -x - 2 \)[/tex] approaches [tex]\( +\infty \)[/tex]. As [tex]\( x \)[/tex] approaches [tex]\(-2 \)[/tex] from the left, [tex]\( -x - 2 \)[/tex] approaches [tex]\( 0 \)[/tex]. Therefore, the range for this interval is [tex]\( (-\infty, -2) \)[/tex].
- For [tex]\( -2 < x < 0 \)[/tex]: The function is [tex]\( f(x) = -x^2 \)[/tex]. The output of a square function is always non-positive, and since there is a negative sign, it will always be non-positive. The maximum value occurs when [tex]\( x \)[/tex] is close to [tex]\( 0 \)[/tex], and the minimum value when [tex]\( x \)[/tex] approaches [tex]\(-2 \)[/tex]. This results in outputs from [tex]\( -4 \)[/tex] (when [tex]\( x = -2 \)[/tex]) to [tex]\( 0 \)[/tex] (when [tex]\( x \)[/tex] is close to 0 but not zero). Hence, the range is [tex]\( (-4, 0) \)[/tex].
- For [tex]\( x \geq 0 \)[/tex]: The function is [tex]\( f(x) = x \)[/tex]. As [tex]\( x \)[/tex] increases to infinity, [tex]\( y = x \)[/tex] increases to infinity. Therefore, the range for this interval is [tex]\( [0, \infty) \)[/tex].
Combining all ranges, we get:
[tex]\[ (-\infty, -2) \cup (0, \infty) \][/tex]
### Final answers:
Domain of [tex]\( f(x) \)[/tex]:
[tex]\[ (-\infty, -2) \cup (-2, \infty) \][/tex]
Range of [tex]\( f(x) \)[/tex]:
[tex]\[ (-\infty, -2) \cup (0, \infty) \][/tex]
There you go! The domain of the function [tex]\( f(x) \)[/tex] is [tex]\( (-\infty, -2) \cup (-2, \infty) \)[/tex] and the range is [tex]\( (-\infty, -2) \cup (0, \infty) \)[/tex].
