How to find the range of piecewise functions?
Understand the Problem
The question is asking how to determine the range of piecewise functions, which involves finding the output values corresponding to the different pieces of the function across their specified domains.
Answer
The final range of the piecewise function should be expressed in interval notation as a combination of the ranges found for each piece.
Answer for screen readers
The range of the piecewise function will depend on the specified pieces and their domains. It can be expressed in interval notation once all calculations are completed.
Steps to Solve
- Identify the Pieces of the Piecewise Function
Start by examining the piecewise function. Each piece has a different expression and domain. Write down each expression and its corresponding domain.
- Determine Output Values for Each Piece
For each piece, evaluate the function over its domain to find the minimum and maximum output values. For example, if one piece is $f(x) = x^2$ and the domain is $[0, 2]$, calculate the values at the endpoints and within the interval, such as:
$$ f(0) = 0^2 = 0 $$
$$ f(2) = 2^2 = 4 $$
- Combine the Ranges of Each Piece
Once you've calculated the ranges (minimum and maximum output values) for each piece, combine these to find the overall range of the piecewise function. Ensure that none of the ranges overlap and consider the union of intervals if necessary.
- Consider Open and Closed Intervals
Be mindful of whether the endpoints of the domains are included (closed) or not included (open), as this affects the final range. For instance, a domain of $[a, b)$ means $a$ is included, while $b$ is not.
- Express the Final Range
Finally, write the combined range in interval notation. For example, if the piecewise function produces the outputs from the intervals $[0, 4]$ and $(5, 7]$, your final range would be:
$$ [0, 4] \cup (5, 7] $$
The range of the piecewise function will depend on the specified pieces and their domains. It can be expressed in interval notation once all calculations are completed.
More Information
Piecewise functions can represent different behaviors in different intervals. Understanding how to find their ranges is crucial in various applications such as optimization problems.
Tips
- Ignoring the domains of each piece can lead to incorrect ranges.
- Forgetting to consider whether endpoints are included or excluded.
- Not combining overlapping ranges correctly.
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