Understanding the Domain and Range of the Function ( y frac{2x - 1}{3x - 2} )

When working with mathematical functions, particularly rational functions like ( y frac{2x - 1}{3x - 2} ), determining the domain and range is crucial. This article walks through the process step-by-step, ensuring a comprehensive understanding of the function's characteristics.

Domain of the Function

The domain of a function includes all the values of ( x ) for which the function is defined. For rational functions, the function is undefined wherever the denominator is zero. Let's analyze the given function ( y frac{2x - 1}{3x - 2} ).

Step-by-Step Analysis for the Domain

Set the denominator to zero to determine where the function is undefined:

[ 3x - 2 0 ]

Solve for ( x ):

[ 3x 2 ] [ x frac{2}{3} ]

Therefore, the function is defined for all real numbers except ( x frac{2}{3} ).

The domain of the function is:

[ text{Domain: } mathbb{R} setminus left{frac{2}{3}right} quad text{or in interval notation: } left(-infty, frac{2}{3}right) cup left(frac{2}{3}, inftyright) ]

Range of the Function

The range of a function is the set of all possible output values. To find the range, we analyze the behavior of the function, particularly the vertical and horizontal asymptotes.

Vertical Asymptote

As ( x ) approaches ( -frac{2}{3} ), the function ( y frac{2x - 1}{3x - 2} ) approaches ( pm infty ). This indicates the presence of a vertical asymptote at ( x -frac{2}{3} ).

Horizontal Asymptote

As ( x ) approaches ( pm infty ), the function behaves like:

[ y frac{2x}{3x} frac{2}{3} ]

Thus, there is a horizontal asymptote at ( y frac{2}{3} ).

Finding Specific ( y ) Values

Rearrange the equation ( y frac{2x - 1}{3x - 2} ) to find if there are any values of ( y ) that cannot be achieved:

[ y(3x - 2) 2x - 1 ]

( 3yx - 2y 2x - 1 )

( 3yx - 2x 2y - 1 )

( x(3y - 2) 1 - 2y )

( x frac{1 - 2y}{3y - 2} )

For the function to be defined, ( 3y - 2 eq 0 ). This means:

[ y eq frac{2}{3} ]

Range of the Function

Since the function can take all real values except ( y frac{2}{3} ), the range is:

[ text{Range: } mathbb{R} setminus left{frac{2}{3}right} quad text{or in interval notation: } left(-infty, frac{2}{3}right) cup left(frac{2}{3}, inftyright) ]

Summary

The domain and range of the function ( y frac{2x - 1}{3x - 2} ) are as follows:

Domain: ( mathbb{R} setminus left{frac{2}{3}right} ) or ( (-infty, frac{2}{3}) cup (frac{2}{3}, infty) ) Range: ( mathbb{R} setminus left{frac{2}{3}right} ) or ( (-infty, frac{2}{3}) cup (frac{2}{3}, infty) )

Understanding these characteristics helps in graphing and analyzing the behavior of the function.