Write The Domain For The Piecewise Function In Interval Notation.$\[ F(x) = \begin{cases} 2x - 5 & \text{for } X \leq -1 \\ -2x^2 & \text{for } X \geq 2 \end{cases} \\]

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Introduction

In mathematics, a piecewise function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain. The domain of a piecewise function is the set of all possible input values for which the function is defined. In this article, we will discuss how to write the domain of a piecewise function in interval notation.

Understanding Piecewise Functions

A piecewise function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain. The general form of a piecewise function is:

f(x)={f1(x)for x∈D1f2(x)for x∈D2⋮fn(x)for x∈Dn{ f(x) = \begin{cases} f_1(x) & \text{for } x \in D_1 \\ f_2(x) & \text{for } x \in D_2 \\ & \vdots \\ f_n(x) & \text{for } x \in D_n \end{cases} }

where f1(x),f2(x),…,fn(x)f_1(x), f_2(x), \ldots, f_n(x) are the sub-functions, and D1,D2,…,DnD_1, D_2, \ldots, D_n are the intervals of the domain.

Domain of a Piecewise Function

The domain of a piecewise function is the set of all possible input values for which the function is defined. In interval notation, the domain of a piecewise function is written as a union of intervals.

Example 1: Domain of a Piecewise Function

Consider the piecewise function:

f(x)={2x−5for x≤−1−2x2for x≥2{ f(x) = \begin{cases} 2x - 5 & \text{for } x \leq -1 \\ -2x^2 & \text{for } x \geq 2 \end{cases} }

To find the domain of this function, we need to consider the intervals of the domain for each sub-function.

For the first sub-function, f(x)=2x−5f(x) = 2x - 5, the domain is x≤−1x \leq -1. This means that the input values for this sub-function are all real numbers less than or equal to -1.

For the second sub-function, f(x)=−2x2f(x) = -2x^2, the domain is x≥2x \geq 2. This means that the input values for this sub-function are all real numbers greater than or equal to 2.

Since the two sub-functions are defined on different intervals, the domain of the piecewise function is the union of these intervals.

Writing the Domain in Interval Notation

In interval notation, the domain of the piecewise function is written as:

(−∞,−1]∪[2,∞){ (-\infty, -1] \cup [2, \infty) }

This means that the domain of the function is all real numbers less than or equal to -1, and all real numbers greater than or equal to 2.

Conclusion

In conclusion, the domain of a piecewise function is the set of all possible input values for which the function is defined. In interval notation, the domain of a piecewise function is written as a union of intervals. By understanding how to write the domain of a piecewise function in interval notation, we can better analyze and work with these types of functions.

Tips and Tricks

  • When writing the domain of a piecewise function in interval notation, make sure to include all possible input values for each sub-function.
  • Use the union symbol (∪) to combine the intervals of the domain.
  • Be careful when writing the domain of a piecewise function, as it is easy to make mistakes.

Common Mistakes

  • Failing to include all possible input values for each sub-function.
  • Using the wrong notation for the union of intervals.
  • Making mistakes when writing the domain of a piecewise function.

Real-World Applications

  • Piecewise functions are used in many real-world applications, such as modeling population growth, temperature changes, and financial transactions.
  • Understanding how to write the domain of a piecewise function in interval notation is essential for working with these types of functions.

Practice Problems

  • Find the domain of the piecewise function: f(x)={x2+1for x≤0x−2for x≥1{ f(x) = \begin{cases} x^2 + 1 & \text{for } x \leq 0 \\ x - 2 & \text{for } x \geq 1 \end{cases} }
  • Write the domain of the piecewise function in interval notation.

Answer Key

  • The domain of the piecewise function is (−∞,0]∪[1,∞)(-\infty, 0] \cup [1, \infty).
  • The domain of the piecewise function is (−∞,0]∪[1,∞)(-\infty, 0] \cup [1, \infty).

References

  • [1] "Calculus" by Michael Spivak
  • [2] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Note: The references provided are for general information and are not specific to the topic of piecewise functions.