Solving Rational Functions: A Comprehensive Guide to Analyzing ( frac{x^2 - 4}{x^2 - 9} )

When approaching a rational function, it is essential to understand the underlying components that define its behavior, such as roots, vertical and horizontal asymptotes, and partial fractions. In this guide, we will explore the function ( f(x) frac{x^2 - 4}{x^2 - 9} ) in detail, following standard mathematical procedures to fully analyze it.

Solving for Roots

The roots of a rational function are the values of (x) for which the numerator equals zero. For the function ( f(x) frac{x^2 - 4}{x^2 - 9} ), we start by solving the equation ( x^2 - 4 0 ) to find the roots:

( x^2 - 4 0 )

( (x 2)(x - 2) 0 )

( x -2 ) or ( x 2 )

Identifying Vertical Asymptotes

The vertical asymptotes of a rational function occur where the denominator equals zero. For the function ( f(x) frac{x^2 - 4}{x^2 - 9} ), we solve the equation ( x^2 - 9 0 ) to find the vertical asymptotes:

( x^2 - 9 0 )

( (x 3)(x - 3) 0 )

( x -3 ) or ( x 3 )

These values of (x) indicate the presence of vertical asymptotes at (x -3) and (x 3).

Determining the Horizontal Asymptote

The horizontal asymptote of a rational function can be found by examining the degrees of the numerator and the denominator. For the function ( f(x) frac{x^2 - 4}{x^2 - 9} ), both the numerator and the denominator have the same degree (2). When the degrees are equal, the horizontal asymptote is given by the ratio of the leading coefficients:

( y frac{1}{1} 1 )

Thus, the horizontal asymptote is (y 1).

Calculating the Y-Intercept

The y-intercept of a function is the point where (x 0). For the function ( f(x) frac{x^2 - 4}{x^2 - 9} ), we find the y-intercept by evaluating ( f(0) ):

( f(0) frac{0^2 - 4}{0^2 - 9} )

( f(0) frac{-4}{-9} )

( f(0) approx 0.4444 )

The y-intercept is approximately (0.4444).

Decomposing into Partial Fractions

For more complex rational functions, it can be helpful to decompose them into partial fractions. Starting with the function ( frac{x^2 - 4}{x^2 - 9} ), we rewrite the denominator as follows:

( x^2 - 9 (x 3)(x - 3) )

We want to express the function in the form:

( frac{x^2 - 4}{(x 3)(x - 3)} frac{A}{x - 3} frac{B}{x 3} )

Multiplying through by the factor ((x - 3)(x 3)) to clear the denominators, we get:

( x^2 - 4 A(x 3) B(x - 3) )

Expanding and equating coefficients, we obtain:

( x^2 - 4 Ax 3A Bx - 3B )

( x^2 - 4 (A B)x (3A - 3B) )

From this, we set up the system of equations:

( A B 1 )

( 3A - 3B -4 )

Solving these equations, we first solve for (A) and (B):

( A B 1 )

( A - B -frac{4}{3} )

Adding the two equations:

( 2A -frac{1}{3} )

( A -frac{1}{6} )

Substituting (A) back into the first equation:

( -frac{1}{6} B 1 )

( B frac{7}{6} )

Therefore, the partial fraction decomposition is:

( frac{x^2 - 4}{x^2 - 9} -frac{1}{6(x - 3)} frac{7}{6(x 3)} )

Conclusion

In conclusion, the rational function ( f(x) frac{x^2 - 4}{x^2 - 9} ) can be analyzed through the examination of its roots, vertical and horizontal asymptotes, y-intercept, and partial fraction decomposition. Understanding these components is crucial for a comprehensive analysis of the function.

Related Keywords

- rational function

- roots

- asymptotes

- partial fractions