In this article, we will explore the range of the function fx (x-2)/(3-x). We will start by defining simplifying our notation and then proceed to calculate the domain and range of the given function. Finally, we will use the concept of inverse functions to determine the range more effectively.

Introduction

The function we are analyzing is fx (x-2)/(3-x). To make our work easier, we introduce a substitution, setting y fx. This allows us to compute the domain of the function in a straightforward manner.

Domain Calculation

First, let's determine the domain of the function by setting the denominator to zero:

Determining the Domain

Since the function is of the form y (x-2)/(3-x), we need to ensure the denominator is not zero:

3 - x 0

Solving for x gives us:

x 3

Thus, the function is undefined at x 3. The domain of the function can be expressed as:

Domain of the Function

The domain is therefore:

-∞, 3 U 3, ∞

Finding the Range Using Inverse Functions

To find the range of the original function, we can use the inverse function approach. This involves swapping the variables in the equation and solving for the new function.

Swapping Variables and Inverting the Function

Let's swap the variables and rewrite the function:

x (y - 2)/(3 - y)

Multiplying both sides by (3 - y) to clear the fraction:

x(3 - y) y - 2

Distributing x on the left side:

3x - xy y - 2

Rearranging the equation to isolate y:

3x - 2 y xy

Factoring out y on the right side:

3x - 2 y(1 x)

Solving for y:

y (3x - 2) / (1 x)

Domain of the Inverse Function

Now we need to determine the domain of the inverse function to find the range of the original function:

The inverse function is y (3x - 2) / (1 x). The domain of this function is where the denominator is not zero:

1 x ≠ 0

Solving for x gives us:

x ≠ -1

The domain of the inverse function is therefore:

-∞, -1 U -1, ∞

Conclusion on the Range

Since the domain of the inverse function is the range of the original function, the range of fx (x-2)/(3-x) is:

-∞, -1 U -1, ∞

Key Takeaways

1. The domain of the function fx (x-2)/(3-x) is -∞, 3 U 3, ∞.

2. By finding the domain of the inverse function y (3x - 2)/(1 x), we can determine the range of the original function.

3. The range of the function is -∞, -1 U -1, ∞.

Related Keywords and Terms

function range, inverse function, domain calculation