Could I please have 1 example of a piecewise-defined function problem, with 2 function pieces and the domain and range of each piece. Thank you
Of course! Here's an example of a piecewise-defined function problem with two function pieces and the domain and range of each piece:
Consider the following piecewise function:
\[ f(x) = \begin{cases}
2x + 1 & \text{if } x < 0 \\
x^2 & \text{if } x \geq 0
\end{cases}
\]
In this example, we have two function pieces:
Piece 1: \( 2x + 1 \)
Domain: \( x < 0 \) (all values of \( x \) that are less than zero)
Range: \( f(x) \) can take any real number value as long as \( x \) is less than zero.
Piece 2: \( x^2 \)
Domain: \( x \geq 0 \) (all values of \( x \) that are greater than or equal to zero)
Range: \( f(x) \) can take any real number value as long as \( x \) is greater than or equal to zero.
To evaluate the function, you would need to consider the value of \( x \) in relation to zero. If \( x \) is less than zero, you would plug the value into the first function piece (2x + 1). If \( x \) is greater than or equal to zero, you would use the second function piece (x^2).
For example, if we want to find the value of \( f(x) \) when \( x = -2 \), we would substitute \( x = -2 \) into the first function piece:
\[ f(-2) = 2(-2) + 1 = -3 \]
Alternatively, if we want to find the value of \( f(x) \) when \( x = 2 \), we would substitute \( x = 2 \) into the second function piece:
\[ f(2) = (2)^2 = 4 \]
I hope this example helps illustrate how to work with a piecewise-defined function! Let me know if there's anything else I can assist you with.