Understanding X and Y Intercepts of Rational Functions: Case Study with ( f(x) frac{5}{x^2 - 64} )

In this article, we will delve into the process of finding the x and y intercepts of a rational function, specifically the function ( f(x) frac{5}{x^2 - 64} ). We will explore how these intercepts are determined and understand the significance of each in the context of the given function.

1. Introduction to Rational Functions

A rational function is defined as a ratio of two polynomials. More specifically, a function ( f(x) ) can be expressed as ( f(x) frac{g(x)}{h(x)} ) where ( g(x) ) and ( h(x) ) are polynomials. In our case, ( f(x) frac{5}{x^2 - 64} ) is a rational function where ( g(x) 5 ) and ( h(x) x^2 - 64 ).

2. Finding the Y-Intercept

The y-intercept occurs at the point where the graph of the function ( f(x) ) crosses the y-axis, which is when ( x 0 ).

Step 1: Set ( x 0 ) and solve for ( y ).

( f(0) frac{5}{0^2 - 64} frac{5}{-64} -frac{5}{64} )

Therefore, the y-intercept is the point ((0, -frac{5}{64})).

3. Finding the X-Intercept

The x-intercept occurs at the point where the graph of the function ( f(x) ) crosses the x-axis, which is when ( f(x) 0 ). A rational function is equal to zero when its numerator is zero and the denominator is non-zero.

Step 1: Set the numerator to zero and solve for ( x ).

The numerator of ( f(x) ) is ( 5 ), which is never zero. Therefore, there are no x-intercepts for this function.

4. Alternative Interpretation

Given the original expression ( 5/x^2 - 64 ), if we interpret it differently as ( frac{5}{x^2} - 64 ), we will explore the intercepts under this interpretation.

4.1 Y-Intercept

The y-intercept would be found by setting ( x 0 ).

( f(0) frac{5}{0^2} - 64 )

This expression is undefined because division by zero is not allowed. Therefore, the y-axis serves as a vertical asymptote, meaning there is no y-intercept.

4.2 X-Intercept

The x-intercepts would be found by solving ( frac{5}{x^2} - 64 0 ).

( frac{5}{x^2} 64 )

( 64x^2 5 )

( x^2 frac{5}{64} )

( x pm frac{sqrt{5}}{8} )

The x-intercepts are ( left( frac{sqrt{5}}{8}, 0 right) ) and ( left( -frac{sqrt{5}}{8}, 0 right) ).

5. Conclusion

In conclusion, the y-intercept of ( f(x) frac{5}{x^2 - 64} ) is at ( (0, -frac{5}{64}) ). There are no x-intercepts for this rational function. However, if interpreted as ( frac{5}{x^2} - 64 ), the y-intercept is undefined and the x-intercepts are ( left( frac{sqrt{5}}{8}, 0 right) ) and ( left( -frac{sqrt{5}}{8}, 0 right) ).

Understanding these key points is crucial for analyzing rational functions and interpreting their graphical behavior.