Determining the Overall Range

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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. When dealing with piecewise functions, it's essential to understand the range of each sub-function and how they interact with each other. In this article, we'll explore the range of a given piecewise function and discuss the different intervals where each sub-function is defined.

The Piecewise Function

The given piecewise function is defined as:

${ f(x) = \begin{cases} 3, & x < 0 \\ x^2 + 2, & 0 \leq x < 2 \\ \frac{1}{2}x + 5, & x \geq 2 \end{cases} }$

Analyzing the Sub-Functions

To determine the range of the piecewise function, we need to analyze each sub-function individually.

Sub-Function 1: $f(x) = 3$ for $x < 0$

This sub-function is a constant function that outputs 3 for all values of x less than 0. Since the output is constant, the range of this sub-function is a single point, which is {3}.

Sub-Function 2: $f(x) = x^2 + 2$ for $0 \leq x < 2$

This sub-function is a quadratic function that outputs the square of x plus 2 for all values of x between 0 and 2 (exclusive). To find the range of this sub-function, we need to consider the minimum and maximum values of the quadratic function.

The minimum value of the quadratic function occurs at x = 0, which is f(0) = 2. The maximum value of the quadratic function occurs at x = 2, which is f(2) = 2^2 + 2 = 6.

Therefore, the range of this sub-function is the interval [2, 6].

Sub-Function 3: $f(x) = \frac{1}{2}x + 5$ for $x \geq 2$

This sub-function is a linear function that outputs half of x plus 5 for all values of x greater than or equal to 2. To find the range of this sub-function, we need to consider the minimum and maximum values of the linear function.

The minimum value of the linear function occurs at x = 2, which is f(2) = 5.5. Since the linear function is increasing, the maximum value of the linear function occurs as x approaches infinity.

Therefore, the range of this sub-function is the interval [5.5, ∞).

Determining the Overall Range

To determine the overall range of the piecewise function, we need to consider the ranges of each sub-function and how they interact with each other.

The range of the first sub-function is a single point, {3}. The range of the second sub-function is the interval [2, 6]. The range of the third sub-function is the interval [5.5, ∞).

Since the piecewise function is defined for all real numbers, the overall range is the union of the ranges of each sub-function. Therefore, the overall range of the piecewise function is the interval [2, 6] ∪ [5.5, ∞).

Conclusion

In conclusion, the range of the given piecewise function is the interval [2, 6] ∪ [5.5, ∞). This interval represents all possible output values of the function for different input values.

Answer

The correct answer is:

A. [2, 6] ∪ [5.5, ∞)

Introduction

In our previous article, we explored the range of a given piecewise function and discussed the different intervals where each sub-function is defined. In this article, we'll answer some frequently asked questions about piecewise functions and their ranges.

Q: What is a piecewise function?

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

Q: Why do we need to consider the ranges of each sub-function?

When dealing with piecewise functions, it's essential to understand the range of each sub-function and how they interact with each other. This is because the overall range of the piecewise function is the union of the ranges of each sub-function.

Q: How do we determine the range of a sub-function?

To determine the range of a sub-function, we need to consider the minimum and maximum values of the function. For example, if we have a quadratic function, we need to find the minimum and maximum values of the function by considering the vertex of the parabola.

Q: What is the difference between the domain and the range of a function?

The domain of a function is the set of all possible input values, while the range of a function is the set of all possible output values.

Q: Can a piecewise function have a single range?

Yes, a piecewise function can have a single range if all the sub-functions have the same range.

Q: How do we determine the overall range of a piecewise function?

To determine the overall range of a piecewise function, we need to consider the ranges of each sub-function and how they interact with each other. The overall range is the union of the ranges of each sub-function.

Q: Can a piecewise function have an infinite range?

Yes, a piecewise function can have an infinite range if one of the sub-functions has an infinite range.

Q: How do we graph a piecewise function?

To graph a piecewise function, we need to graph each sub-function separately and then combine them to form the overall graph.

Q: Can a piecewise function be continuous?

Yes, a piecewise function can be continuous if the sub-functions are continuous and the intervals where they are defined are adjacent.

Q: How do we find the derivative of a piecewise function?

To find the derivative of a piecewise function, we need to find the derivative of each sub-function separately and then combine them to form the overall derivative.

Q: Can a piecewise function be used to model real-world phenomena?

Yes, piecewise functions can be used to model real-world phenomena that have different behaviors in different intervals.

Conclusion

In conclusion, piecewise functions are a powerful tool for modeling real-world phenomena that have different behaviors in different intervals. By understanding the range of each sub-function and how they interact with each other, we can determine the overall range of the piecewise function.

Frequently Asked Questions

Q: What is a piecewise function?
A: A piecewise function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain.
Q: Why do we need to consider the ranges of each sub-function?
A: We need to consider the ranges of each sub-function to determine the overall range of the piecewise function.
Q: How do we determine the range of a sub-function?
A: We determine the range of a sub-function by considering the minimum and maximum values of the function.
Q: Can a piecewise function have a single range?
A: Yes, a piecewise function can have a single range if all the sub-functions have the same range.

Glossary

Domain: The set of all possible input values of a function.
Range: The set of all possible output values of a function.
Sub-function: A function that is defined on a specific interval of the domain.
Piecewise function: A function that is defined by multiple sub-functions, each applied to a specific interval of the domain.

References

[1] "Piecewise Functions" by Math Open Reference
[2] "Range of a Function" by Khan Academy
[3] "Piecewise Functions" by Wolfram MathWorld