Understanding the Range of ( f(x) frac{1}{x - sqrt{x^2}} )

In this article, we will explore the range of the given function ( f(x) frac{1}{x - sqrt{x^2}} ) and delve into the necessary steps and considerations for determining its domain and limits.

Introduction to the Function

Given the function:

( f(x) frac{1}{x - sqrt{x^2}} )

The first step in determining the range is to understand the domain of the function. This involves identifying values of x for which the function is defined.

Domain of the Function

For the function to be defined, the denominator must not be zero:

( x - sqrt{x^2} eq 0 )

Let's solve this inequality step by step:

Step 1: Square both sides

 ( x - sqrt{x^2}  0 )( sqrt{x^2}  x )

Since (sqrt{x^2} |x|), this simplifies to:

 ( |x|  x )

For ( |x| x ), x must be non-negative. However, we also have to consider other constraints that arise from the original expression.

Now, let's re-examine (x - sqrt{x^2}):

(x - sqrt{x^2} x - |x|)

When (x geq 0), (|x| x), so (x - x 0), and the denominator becomes zero, making the function undefined. Therefore, (x) cannot be zero or any positive value.

When (x

Simplifying the Domain

Therefore, the domain of (f(x)) is:

( x geq -2 ) and ( x eq 2 )

Thus, the domain can be expressed as:

 text{Domain: } [-2, 2) cup (2, infty)

Analysis of the Function for the Computed Domain

To find the range, we need to analyze the behavior of (f(x)) within its domain.

Checking for Finites and Limits

When (x -2), the function evaluates to:

( f(-2) frac{1}{-2 - sqrt{4}} frac{1}{-2 - 2} frac{1}{-4} -frac{1}{2} )

When approaching (x 2), both from the left and right, the expression (x - sqrt{x^2}) tends to zero from negative and positive values respectively, making the function approach negative infinity from the left and positive infinity from the right:

 displaystyle lim_{x to 2^-} f(x)  -inftydisplaystyle lim_{x to 2^ } f(x)   infty

As (x to infty), the function approaches zero:

 displaystyle lim_{x to infty} f(x)  0

Additionally, we can find specific values to understand the range more clearly. For example, solving for (f(x) 0), we find that:

 f(x)  0frac{1}{x - sqrt{x^2}}  0

This shows no real (x) satisfies this equation. However, examining a value like (x -frac{7}{4}) can give us a specific range point:

( fleft(-frac{7}{4}right) frac{1}{-frac{7}{4} - sqrt{left(-frac{7}{4}right)^2}} -frac{4}{9} )

Range Determination

The value at (x -frac{7}{4}) is a critical point. By examining the function within the domain, we determine that:

The range is:

 left(-infty, -frac{4}{9}right] cup [0, infty)

This indicates that (f(x)) can take any value less than or equal to (-frac{4}{9}) and any value greater than or equal to zero, with a gap at (-frac{4}{9}) due to the critical value found and the behavior at (x 2).

Conclusion

In summary, the range of the function ( f(x) frac{1}{x - sqrt{x^2}} ) is:

 left(-infty, -frac{4}{9}right] cup [0, infty)

This detailed analysis shows that the function has a specific domain and range, demonstrating the interplay of constraints and behavior within the given constraints.