Understanding the Range of the Function fx 1 - x - 2

In this article, we will explore the range of the function fx 1 - x - 2. We will start by examining the domain of the function and then delve into the range, identifying any maximum or minimum values, and understanding the behavior of the function around a specific point of interest, specifically when x 2.

The Domain of the Function

The domain of a function indicates the set of all possible input values, or x-values, for which the function is defined. For the function fx 1 - x - 2, there are no restrictions on the input values. Thus, the domain is:

Domain of fx is {x -∞

The Range of the Function

The range of a function is the set of all possible output values, or y-values, that the function can produce. For the function fx 1 - x - 2, we need to analyze the behavior of the function to determine its range.

Exploring the Range

When x 2, we find:

fx 1 - x - 2 1 - 2 - 2 -3

However, this point doesn't seem to be particularly significant for determining the range. Instead, let's explore the function around other values of x.

When x > 2, the values within the absolute value bars increase, and the function decreases. For example, when x 3:

fx 1 - 3 - 2 -4

With x , the values within the absolute value bars are negative and decreasing. Removing the absolute value bars, they become positive and increasing, and the function decreases. To understand this better, let's calculate fx when x 1:

fx 1 - 1 - 2 -2

From these calculations, it appears that the function reaches its maximum value at x 2, and from there, it decreases. Therefore, the maximum value of the function fx 1 - x - 2 is fx -3.

Thus, the range of the function is:

Range {fx | fx less than or equal to -3}

Comparison with Other Functions

Let's compare the function fx 1 - x - 2 with some other related functions:

1. The function hx x

The range of hx x is [0, ∞). This is because x can take any non-negative value.

2. The function gx x - 2

The range of gx x - 2 is also [0, ∞). This is just a shift of the x-axis by two units to the left.

3. The function px -x - 2

The range of px -x - 2 is [-∞, 0]. This is because the function decreases as x increases, and it reaches its minimum value of -2 at x 0.

4. The function fx -x - 2 1 1 - x - 2

The range of fx 1 - x - 2 is [-∞, -3]. This is because the function is a linear transformation of px -x - 2, shifted upwards by 1 unit. Therefore, the minimum value of the function fx remains -3, but the range extends to negative infinity.

Conclusion

In summary, the function fx 1 - x - 2 has a range of {fx | fx less than or equal to -3}. It reaches its maximum value at x 2, and from there, it decreases as x moves away from 2 in both directions. Understanding the domain and range of a function is crucial for analyzing its behavior and identifying key points.