Exploring the Range of Functions: Understanding (frac{1}{x^2-1}) and Related Concepts

In the field of mathematics, the range of a function is a fundamental concept that describes all the possible output values of the function. In this article, we will explore the range of the function ( f(x) frac{1}{x^2-1} ) and related concepts such as quadratic equations and domain analysis. We will also discuss how to determine the domain and range of the function ( g(x) frac{x}{x^2-1} ).

The Range of ( f(x) frac{1}{x^2-1} )

To determine the range of the function ( f(x) frac{1}{x^2-1} ), we will analyze its behavior through careful domain analysis and the behavior of the denominator.

Domain

The function ( f(x) frac{1}{x^2-1} ) is defined for all real numbers ( x ) because ( x^2-1 ) is always positive. This is because:

For ( x 0 ), ( x^2-1 -1 ), which means the denominator is never zero. As ( x ) increases or decreases, ( x^2-1 ) increases without bound.

Behavior of the Denominator

The expression ( x^2-1 ) is always greater than or equal to 1 for all real ( x ): The minimum value of ( x^2-1 ) occurs at ( x 0 ), where ( x^2-1 -1 ). As ( x ) moves away from 0 (both positively and negatively), ( x^2-1 ) increases without bound.

Function Behavior

Given the above, we can determine the behavior of the function ( f(x) frac{1}{x^2-1} ):

At ( x 0 ), ( f(0) frac{1}{0^2-1} -1 ). As ( x ) approaches infinity, ( f(x) ) approaches 0.

Range

The range of the function ( f(x) frac{1}{x^2-1} ) can be determined by analyzing the function's limits and behavior:

Since ( f(x) ) achieves a maximum value of 1 when ( x 0 ) and approaches 0 as ( x ) increases, the range of the function is:

[ [0,1] ]

Exploring ( y frac{x}{x^2-1} )

Let's consider the function ( y frac{x}{x^2-1} ) and understand how to find its range.

Solving for ( x ) in terms of ( y ), we have:

[ y(x^2-1) x ]

Rewriting this expression as a quadratic equation in ( y ), we get:

[ yx^2 - xy - 1 0 ]

Solving this quadratic equation for ( x ), we have:

[ x frac{y pm sqrt{y^2 4y^2}}{2y} frac{1 pm sqrt{1 4y^2}}{2} ]

Given that ( y ) is always greater than 0, the domain of ( f(x) frac{1}{x^2-1} ) is the range of ( g(x) frac{x}{x^2-1} ).

Range of ( y frac{x}{x^2-1} )

From the above analysis, we can deduce that the range of ( y frac{x}{x^2-1} ) is:

[ left[ -frac{1}{2}, frac{1}{2} right] ]

Determining Domain and Range for ( f(x) frac{x-1}{x 1} )

To find the domain and range of the function ( f(x) frac{x-1}{x 1} ), we follow a systematic approach:

Domain

The domain of the function is determined by the values of ( x ) that make the denominator zero, which means:

[ x 1 eq 0 Rightarrow x eq -1 ]

Thus, the domain of ( f(x) frac{x-1}{x 1} ) is:

[ (-infty, -1) cup (-1, infty) ]

Range

To determine the range, we analyze the behavior of the function as ( x ) approaches ( pm infty ) and the asymptotic behavior:

As ( x ) approaches ( pm infty ), ( f(x) ) approaches 1.

The function is undefined at ( x -1 ), and as ( x ) approaches ( -1 ), ( f(x) ) tends to ( pm infty ).

Therefore, the range of ( f(x) frac{x-1}{x 1} ) is:

[ (-infty, 1) cup (1, infty) ]

Conclusion

In this article, we have explored the range of the function ( f(x) frac{1}{x^2-1} ) and the related function ( g(x) frac{x}{x^2-1} ). We have also discussed how to determine the domain and range of the function ( f(x) frac{x-1}{x 1} ).

The key takeaways are:

Understanding the domain of a function is crucial for determining its range. Quadratic equations can be used to find the range of a function. Asymptotic behavior helps in determining the range of rational functions.