## Write Domain and Range of Piecewise Function – An Overview

### Introduction:

Hello Friends ReviewHost.Plafon.id! Today, we will discuss a very important concept in mathematics – Domain and Range. These two terms are extremely crucial when it comes to studying functions, and in particular, Piecewise Functions.

In this article, we will discuss what domain and range are, and how they are calculated for Piecewise Functions. We will use some examples to illustrate the methods used for finding the domain and range of Piecewise Functions.

### What is Domain?

Domain refers to the set of all possible input values (also called the independent variable) for which the function can produce valid output values. In other words, it is the set of all permissible values that can be used as input for the function.

For instance, consider the function f(x) = 1/x. Here, the denominator x cannot be equal to zero as it leads to division by zero, which is undefined. Therefore, the domain of the function f(x) = 1/x is all real numbers, excluding zero.

In case of Piecewise Functions, the domain is the set of all input values (or independent variable) that are allowed for each piece of the function.

### What is Range?

Range refers to the set of all possible output values (also called the dependent variable) that are produced by the function, given all the possible input values. In other words, it is the set of all permissible values that can be output by the function.

For instance, consider the function f(x) = x^2. Here, the range of the function is all non-negative real numbers, since the square of any real number is always non-negative.

In case of Piecewise Functions, the range is the set of all output values that are allowed for each piece of the function.

### How to Find Domain and Range of Piecewise Functions?

To find the domain and range of a Piecewise Function, we need to evaluate each piece of the function separately. To do so, we identify the intervals where each piece of the function is defined, and then compute the domain and range individually for each piece.

Let us consider a few examples to illustrate the method for finding the domain and range of Piecewise Functions.

### Example 1:

Suppose we have the following Piecewise Function:

f(x) =

-2x + 4, if x <= 3 x^2 - 5, if x > 3

We need to find the domain and range of this function.

To find the domain of the function, we need to find the set of all input values (or independent variable) for which the function is defined. In this case, the function is defined in two pieces – one for the interval x <= 3 and the other for x > 3.

For the first piece of the function (-2x + 4), the domain is all real numbers less than or equal to 3, since the function is defined only for x values less than or equal to 3.

For the second piece of the function (x^2 – 5), the domain is all real numbers greater than 3, since the function is defined only for x values greater than 3.

Thus, the domain of the Piecewise Function is the union of these two intervals, which is (-infinity, 3] U (3, infinity).

To find the range of the function, we need to evaluate each piece of the function separately.

For the first piece of the function (-2x + 4), since the coefficient of x is negative, the function decreases as x increases. Therefore, the largest value that the function can take in this interval is f(3) = -2(3) + 4 = -2. Hence, the range of this piece of the function is (-infinity, -2].

For the second piece of the function (x^2 – 5), since the function is a quadratic function with a positive leading coefficient, the function increases as x increases. Therefore, the smallest value that the function can take in this interval is f(3) = 3^2 – 5 = 4. Hence, the range of this piece of the function is [4, infinity).

Thus, the range of the Piecewise Function is the union of these two intervals, which is (-infinity, -2] U [4, infinity).

### Example 2:

Suppose we have the following Piecewise Function:

f(x) =

x + 5, if x < -3 -x^2 + 2x + 11, if -3 <= x < 2 3, if x >= 2

We need to find the domain and range of this function.

To find the domain of the function, we need to find the set of all input values (or independent variable) for which the function is defined. In this case, the function is defined in three pieces – one for the interval x < -3, the second for -3 <= x < 2, and the third for x >= 2.

For the first piece of the function (x + 5), the domain is all real numbers less than -3, since the function is defined only for x values less than -3.

For the second piece of the function (-x^2 + 2x + 11), the domain is the interval [-3, 2) since the function is defined only for x values between -3 and 2, but not including 2.

For the third piece of the function (3), the domain is all real numbers greater than or equal to 2, since the function is defined only for x values greater than or equal to 2.

Thus, the domain of the Piecewise Function is the union of these three intervals, which is (-infinity, -3) U [-3, 2) U [2, infinity).

To find the range of the function, we need to evaluate each piece of the function separately.

For the first piece of the function (x + 5), since the function is a linear function with a positive slope, the function increases as x increases. Therefore, the smallest value that the function can take in this interval is f(-3) = -3 + 5 = 2. Hence, the range of this piece of the function is [2, infinity).

For the second piece of the function (-x^2 + 2x + 11), since the coefficient of x^2 is negative, the function decreases as x increases in the interval [-3, 2). Also, since the vertex of the parabola is at x = 0.5 (which lies in the interval [-3, 2)), the maximum value of the function occurs at x = 0.5. Therefore, the maximum value of the function in this interval is f(0.5) = -0.5^2 + 2(0.5) + 11 = 11.25. Hence, the range of this piece of the function is (-infinity, 11.25].

For the third piece of the function (3), the range is simply the singleton set 3, since the function is a constant function.

Thus, the range of the Piecewise Function is the union of these three intervals, which is [2, infinity) U (-infinity, 11.25] U 3.

### Conclusion:

Domain and Range are extremely important concepts in mathematics, particularly when it comes to studying functions. In case of Piecewise Functions, these concepts become even more crucial, since the domain and range need to be evaluated for each piece of the function individually.

In this article, we discussed the definition of domain and range, and how to find the domain and range of Piecewise Functions. We illustrated the methods used to find the domain and range with the help of two examples.

We hope that this article provides you with a better understanding of domain and range, and how to calculate them for Piecewise Functions.

That’s all for now! See you in our next interesting article.

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