Questions: Let f(x) = (3x+2)/(6x-3) Find: 4) X-value of any holes 5) Vertical asymptotes at x= 6) Horizontal asymptote at y= 7) Oblique asymptote at y=

Let f(x) = (3x+2)/(6x-3) Find: 4) X-value of any holes 5) Vertical asymptotes at x= 6) Horizontal asymptote at y= 7) Oblique asymptote at y=
Transcript text: Let $f(x)=\frac{3 x+2}{6 x-3}$ Find: 4) X-value of any holes $\square$ 5) Vertical asymptotes at $x=$ $\square$ 6) Horizontal asymptote at $y=$ $\square$ 7) Oblique asymptote at $y=$ $\square$
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Solution

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Solution Steps

Step 1: Finding Holes

Holes exist when there is a factor common to both the numerator and denominator. Since 3_x + 2 and 6_x - 3 share no common factors, there are no holes.

Step 2: Finding Vertical Asymptotes

Vertical asymptotes are found when the denominator is equal to zero. In this case, we set 6x - 3 = 0. Solving for x, we find x = 1/2.

Step 3: Finding the Horizontal Asymptote

Horizontal asymptotes are determined by comparing the degrees of the numerator and denominator. In this case, both are degree 1. The horizontal asymptote is thus the ratio of the leading coefficients, which is y = 3/6 or y = 1/2.

Final Answer:

  1. None
  2. x = 1/2
  3. y = 1/2
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