To find the asymptotes of the given function [tex]\( f(x) = \frac{2ax - 3}{b - 3x} \)[/tex], we need to consider both the vertical and horizontal asymptotes.
### Vertical Asymptote
A vertical asymptote occurs when the denominator of the function equals zero, causing the function to become undefined. To find this, we set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ b - 3x = 0 \][/tex]
Solving for [tex]\( x \)[/tex], we get:
[tex]\[ 3x = b \][/tex]
[tex]\[ x = \frac{b}{3} \][/tex]
Thus, the vertical asymptote is:
[tex]\[ x = \frac{b}{3} \][/tex]
### Horizontal Asymptote
To find the horizontal asymptote of a rational function, we compare the degrees of the numerator and the denominator. Here, both the numerator ( [tex]\( 2ax - 3 \)[/tex] ) and the denominator ( [tex]\( b - 3x \)[/tex] ) are linear polynomials, which means both have the same degree (degree 1).
When the degrees of the numerator and the denominator are the same, the horizontal asymptote is determined by the ratio of the leading coefficients of the numerator and the denominator.
- The leading coefficient of the numerator [tex]\( 2ax - 3 \)[/tex] is [tex]\( 2a \)[/tex].
- The leading coefficient of the denominator [tex]\( b - 3x \)[/tex] is [tex]\( -3 \)[/tex] (keeping in mind the negative sign with [tex]\( -3x \)[/tex]).
Therefore, the horizontal asymptote is:
[tex]\[ y = \frac{2a}{-3} \][/tex]
[tex]\[ y = -\frac{2a}{3} \][/tex]
### Summary
- Vertical asymptote [tex]\( x = \frac{b}{3} \)[/tex]
- Horizontal asymptote [tex]\( y = -\frac{2a}{3} \)[/tex]
