Write A Function For The Piecewise Function Given Below. Then State The Domain And Range
The Correct Answer and Explanation is :
The piecewise function you’ve provided can be written as follows:
[
f(x) =
\begin{cases}
-x & \text{if } x < 0 \ x^2 & \text{if } 0 \leq x \leq 2 \ 2x – 3 & \text{if } x > 2
\end{cases}
]
Domain:
The domain of the piecewise function is the set of all real numbers, since each piece of the function is defined over specific intervals that together cover all real numbers. Thus, the domain is ( (-\infty, \infty) ).
Range:
- For the first part of the function ( f(x) = -x ) where ( x < 0 ), the output will be positive values for negative ( x ), so the range of this segment is ( (0, \infty) ).
- The second part ( f(x) = x^2 ) where ( 0 \leq x \leq 2 ) will produce values from ( 0 ) to ( 4 ), so the range of this segment is ( [0, 4] ).
- The third part ( f(x) = 2x – 3 ) where ( x > 2 ) will produce values greater than ( 1 ), as when ( x = 2 ), ( f(x) = 1 ), and as ( x ) increases, so does ( f(x) ).
Therefore, the range of the entire function is ( [0, \infty) ).
This function is continuous, but it’s piecewise defined with different expressions based on the value of ( x ), resulting in distinct behaviors in different intervals.
If you need further elaboration or more specific details, feel free to ask!
