Question
\[ f(x)=\frac{2 x+6}{x-3} \] yont: a-int: Hor asymp: \( y=2 \) Ver asymp: \( x=3 \) Dimain: Range:
Ask by John Schmidt. in the United States
Jan 30,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The horizontal asymptote is \( y=2 \).
Solution
Find the horizontal asymptote of the function \( f(x)=\frac{2x+6}{x-3} \).
Function by following steps:
- step0: Find the horizontal asymptotes:
\(f\left(x\right)=\frac{2x+6}{x-3}\)
- step1: Evaluate the limits \(\lim _{x\rightarrow +\infty}\left(f\left(x\right)\right)\) and \(\lim _{x\rightarrow -\infty}\left(f\left(x\right)\right):\)
\(\begin{align}&\lim _{x\rightarrow +\infty}\left(\frac{2x+6}{x-3}\right)\\&\lim _{x\rightarrow -\infty}\left(\frac{2x+6}{x-3}\right)\end{align}\)
- step2: Calculate:
\(\begin{align}&2\\&2\end{align}\)
- step3: The finite values are horizontal asymptotes:
\(\begin{align}&f\left(x\right)=2\end{align}\)
The horizontal asymptote of the function \( f(x)=\frac{2x+6}{x-3} \) is \( y=2 \).
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution Extra Insights
To find the domain of the function \( f(x)=\frac{2 x+6}{x-3} \), we need to determine where the function is defined. It is undefined at \( x=3 \) since that makes the denominator zero. Therefore, the domain can be expressed as all real numbers except \( x=3 \): \( \text{Domain: } (-\infty, 3) \cup (3, \infty) \). As for the range, the horizontal asymptote \( y=2 \) indicates that as \( x \) approaches positive or negative infinity, \( f(x) \) approaches 2, but never actually reaches it. Since there are no restrictions on \( y \) except this asymptote, the range is \( \text{Range: } (-\infty, 2) \cup (2, \infty) \).
