What are the asymptotes for #f(x)=[(5x+3)/(2x-3)]+1#?

Question

Serenity Jones · Accepted Answer

#3/2#

You have to ask yourself: what does X have to be for the function to be dividing by 0?

#2x-3=0# #2x=3# #x=3/2#

So when #x# is #3/2#, The function will contain a divide by zero error, and that is the asymptote. The "#+1#" at the end will not affect this, but will to the roots/zeros/x-intercepts.

Theodore Amidon · Answer

undefined To find the asymptotes of the function $ f(x) = \frac{5x+3}{2x-3} + 1 $, we need to examine the behavior of the function as $ x $ approaches certain values.

Vertical asymptotes occur when the denominator of the rational function becomes zero, leading to undefined values. In this case, the denominator $ 2x - 3 $ becomes zero when $ x = \frac{3}{2} $. Therefore, the vertical asymptote is $ x = \frac{3}{2} $.

Horizontal asymptotes describe the behavior of the function as $ x $ approaches positive or negative infinity. To find horizontal asymptotes, we examine the degrees of the numerator and denominator. In this case, both the numerator and denominator have degrees of 1. When the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients. Thus, the horizontal asymptote is $ y = \frac{5}{2} $.

Therefore, the asymptotes for the function $ f(x) = \frac{5x+3}{2x-3} + 1 $ are a vertical asymptote at $ x = \frac{3}{2} $ and a horizontal asymptote at $ y = \frac{5}{2} $.