Exploring the Function f(x) (xa - x) / (xb - x)

In the realm of algebra, functions offer a powerful tool to understand the relationship between variables. An intriguing question often arises from algebraic expressions: what is the answer to [xa] - [x] f [x] [xb] - [x] [xb]? Let's delve into the details to unravel the complexity of this function.

Understanding the Function

Let's start by simplifying the given expression:

[ [xa] - [x] f [x] [xb] - [x] [xb] ]

By subtracting [x] from both sides, we get:

[ [xa] - [x] f [xb] - [x] [xb] ]

Further simplification leads to:

[ [xa] f [xb] ]

Dividing both sides by [xb], we obtain:

[ f(x) frac{xa}{xb} ]

Function Characteristics

The function ( f(x) frac{xa}{xb} ) is a simple linear fraction. It is a function of one variable, x, and includes two constants, ( a ) and ( b ). Notably, this function is not defined when the denominator becomes zero, i.e., ( x -b ). This restriction means that the function has a singularity at ( x -b ).

Graphical Representation

Let's explore the graphical representation of this function. By substituting different values of ( a ) and ( b ), we can observe various behaviors of the function.

Example 1: When ( a b ), the function simplifies to:

[ f(x) frac{xb}{xb} 1 ] for ( x eq -b )

In this case, the function is the horizontal line ( y 1 ), with an exception at the point ( (-b, 1) ). This is because ( f(x) ) is undefined at this point due to the zero in the denominator.

Example 2: If ( a eq b ), the function ( f(x) frac{xa}{xb} ) remains a linear fraction. The value of ( f(x) ) will change based on the ratio of ( a ) and ( b ).

Graphing the Function

To visualize this function, we can use graphing tools like Desmos. Here's how to input the function:

Desmos Input: ( f(x) frac{xa}{xb} )

Navigate to Desmos () and enter the function. You can set values for ( a ) and ( b ) to see the different behaviors of the function. For example:

For ( a 2 ) and ( b 1 ): ( f(x) frac{2x}{x - 1} ) For ( a 3 ) and ( b 4 ): ( f(x) frac{3x}{4x - 4} )

These examples will help you better understand the nature of the function and its graph.

Conclusion

The function ( f(x) frac{xa}{xb} ) is a fundamental algebraic expression that offers insights into the behavior of linear fractions. By simplifying the given expression, we can derive a clear understanding of its properties and visualize its graph using tools like Desmos. The function's singularity at ( x -b ) and its linear behavior when ( a b ) are key aspects to explore.

If you have any further questions or need more detailed analysis, feel free to explore the resources available on Desmos and other mathematical platforms.

Related Keywords

function, algebra, graph