Answer :
To determine which function has a graph with a horizontal asymptote at [tex]\( y=3 \)[/tex], a vertical asymptote at [tex]\( x=1 \)[/tex], and an [tex]\( x \)[/tex]-intercept at 2, let's analyze each option step by step.
### 1. Horizontal Asymptote
The horizontal asymptote of a rational function [tex]\( f(x) = \frac{a_n x^n + \cdots + a_0}{b_m x^m + \cdots + b_0} \)[/tex] depends on the degrees of the numerator (n) and the denominator (m):
- If [tex]\( n < m \)[/tex], the horizontal asymptote is [tex]\( y=0 \)[/tex].
- If [tex]\( n = m \)[/tex], the horizontal asymptote is [tex]\( y = \frac{a_n}{b_m} \)[/tex].
- If [tex]\( n > m \)[/tex], there is no horizontal asymptote (the function may have an oblique asymptote).
We are looking for a horizontal asymptote at [tex]\( y=3 \)[/tex], which implies that the degrees of the numerator and the denominator are equal, and the leading coefficient ratio is 3.
### 2. Vertical Asymptote
A vertical asymptote occurs where the denominator of the rational function is zero, as long as those points do not also cancel with zeros in the numerator:
- We need the vertical asymptote at [tex]\( x=1 \)[/tex], meaning the denominator should have a factor of [tex]\( (x-1) \)[/tex].
### 3. X-Intercept
An [tex]\( x \)[/tex]-intercept occurs where the numerator is zero:
- We need the [tex]\( x \)[/tex]-intercept at [tex]\( x=2 \)[/tex], meaning the numerator should have a factor of [tex]\( (x-2) \)[/tex].
### Analysis of Each Function
#### Option 1: [tex]\( f(x)=\frac{x-1}{3(x-2)} \)[/tex]
- Horizontal Asymptote: The degrees are equal, and the leading coefficient ratio is [tex]\( \frac{1}{3} \)[/tex]. This suggests a horizontal asymptote at [tex]\( y = 1/3 \)[/tex], not [tex]\( y=3 \)[/tex].
- Vertical Asymptote: [tex]\( x-2 = 0 \)[/tex] gives [tex]\( x=2 \)[/tex], which is not at [tex]\( x=1 \)[/tex].
- X-Intercept: [tex]\( x-1 = 0 \)[/tex] gives [tex]\( x=1 \)[/tex], not at [tex]\( x=2 \)[/tex].
This function does not meet any of the criteria.
#### Option 2: [tex]\( f(x)=\frac{3(x-1)}{x-2} \)[/tex]
- Horizontal Asymptote: The degrees are equal, and the leading coefficient ratio is [tex]\( 3 \)[/tex]. This suggests a horizontal asymptote at [tex]\( y = 3 \)[/tex].
- Vertical Asymptote: [tex]\( x-2 = 0 \)[/tex] gives [tex]\( x=2 \)[/tex], which is not at [tex]\( x=1 \)[/tex].
- X-Intercept: [tex]\( 3(x-1) = 0 \)[/tex] gives [tex]\( x=1 \)[/tex], not at [tex]\( x=2 \)[/tex].
This function does not meet the criteria for the vertical asymptote or the [tex]\( x \)[/tex]-intercept.
#### Option 3: [tex]\( f(x)=\frac{x-2}{3(x-1)} \)[/tex]
- Horizontal Asymptote: The degrees are equal, and the leading coefficient ratio is [tex]\( \frac{1}{3} \)[/tex]. This suggests a horizontal asymptote at [tex]\( y = 1/3 \)[/tex], not [tex]\( y=3 \)[/tex].
- Vertical Asymptote: [tex]\( 3(x-1) = 0 \)[/tex] gives [tex]\( x=1 \)[/tex], which is correct.
- X-Intercept: [tex]\( x-2 = 0 \)[/tex] gives [tex]\( x=2 \)[/tex], also correct.
This function meets the criteria for the vertical asymptote and the [tex]\( x \)[/tex]-intercept but not for the horizontal asymptote.
#### Option 4: [tex]\( f(x)=\frac{3(x-2)}{x-1} \)[/tex]
- Horizontal Asymptote: The degrees are equal, and the leading coefficient ratio is [tex]\( 3 \)[/tex]. This suggests a horizontal asymptote at [tex]\( y = 3 \)[/tex], correct.
- Vertical Asymptote: [tex]\( x-1 = 0 \)[/tex] gives [tex]\( x=1 \)[/tex], correct.
- X-Intercept: [tex]\( 3(x-2) = 0 \)[/tex] gives [tex]\( x=2 \)[/tex], correct.
This function meets all the required criteria.
### Conclusion
The function that satisfies a horizontal asymptote at [tex]\( y=3 \)[/tex], a vertical asymptote at [tex]\( x=1 \)[/tex], and an [tex]\( x \)[/tex]-intercept at [tex]\( x=2 \)[/tex] is:
[tex]\[ f(x) = \frac{3(x-2)}{x-1} \][/tex]
So, the correct answer is:
[tex]\[ \boxed{4} \][/tex]
### 1. Horizontal Asymptote
The horizontal asymptote of a rational function [tex]\( f(x) = \frac{a_n x^n + \cdots + a_0}{b_m x^m + \cdots + b_0} \)[/tex] depends on the degrees of the numerator (n) and the denominator (m):
- If [tex]\( n < m \)[/tex], the horizontal asymptote is [tex]\( y=0 \)[/tex].
- If [tex]\( n = m \)[/tex], the horizontal asymptote is [tex]\( y = \frac{a_n}{b_m} \)[/tex].
- If [tex]\( n > m \)[/tex], there is no horizontal asymptote (the function may have an oblique asymptote).
We are looking for a horizontal asymptote at [tex]\( y=3 \)[/tex], which implies that the degrees of the numerator and the denominator are equal, and the leading coefficient ratio is 3.
### 2. Vertical Asymptote
A vertical asymptote occurs where the denominator of the rational function is zero, as long as those points do not also cancel with zeros in the numerator:
- We need the vertical asymptote at [tex]\( x=1 \)[/tex], meaning the denominator should have a factor of [tex]\( (x-1) \)[/tex].
### 3. X-Intercept
An [tex]\( x \)[/tex]-intercept occurs where the numerator is zero:
- We need the [tex]\( x \)[/tex]-intercept at [tex]\( x=2 \)[/tex], meaning the numerator should have a factor of [tex]\( (x-2) \)[/tex].
### Analysis of Each Function
#### Option 1: [tex]\( f(x)=\frac{x-1}{3(x-2)} \)[/tex]
- Horizontal Asymptote: The degrees are equal, and the leading coefficient ratio is [tex]\( \frac{1}{3} \)[/tex]. This suggests a horizontal asymptote at [tex]\( y = 1/3 \)[/tex], not [tex]\( y=3 \)[/tex].
- Vertical Asymptote: [tex]\( x-2 = 0 \)[/tex] gives [tex]\( x=2 \)[/tex], which is not at [tex]\( x=1 \)[/tex].
- X-Intercept: [tex]\( x-1 = 0 \)[/tex] gives [tex]\( x=1 \)[/tex], not at [tex]\( x=2 \)[/tex].
This function does not meet any of the criteria.
#### Option 2: [tex]\( f(x)=\frac{3(x-1)}{x-2} \)[/tex]
- Horizontal Asymptote: The degrees are equal, and the leading coefficient ratio is [tex]\( 3 \)[/tex]. This suggests a horizontal asymptote at [tex]\( y = 3 \)[/tex].
- Vertical Asymptote: [tex]\( x-2 = 0 \)[/tex] gives [tex]\( x=2 \)[/tex], which is not at [tex]\( x=1 \)[/tex].
- X-Intercept: [tex]\( 3(x-1) = 0 \)[/tex] gives [tex]\( x=1 \)[/tex], not at [tex]\( x=2 \)[/tex].
This function does not meet the criteria for the vertical asymptote or the [tex]\( x \)[/tex]-intercept.
#### Option 3: [tex]\( f(x)=\frac{x-2}{3(x-1)} \)[/tex]
- Horizontal Asymptote: The degrees are equal, and the leading coefficient ratio is [tex]\( \frac{1}{3} \)[/tex]. This suggests a horizontal asymptote at [tex]\( y = 1/3 \)[/tex], not [tex]\( y=3 \)[/tex].
- Vertical Asymptote: [tex]\( 3(x-1) = 0 \)[/tex] gives [tex]\( x=1 \)[/tex], which is correct.
- X-Intercept: [tex]\( x-2 = 0 \)[/tex] gives [tex]\( x=2 \)[/tex], also correct.
This function meets the criteria for the vertical asymptote and the [tex]\( x \)[/tex]-intercept but not for the horizontal asymptote.
#### Option 4: [tex]\( f(x)=\frac{3(x-2)}{x-1} \)[/tex]
- Horizontal Asymptote: The degrees are equal, and the leading coefficient ratio is [tex]\( 3 \)[/tex]. This suggests a horizontal asymptote at [tex]\( y = 3 \)[/tex], correct.
- Vertical Asymptote: [tex]\( x-1 = 0 \)[/tex] gives [tex]\( x=1 \)[/tex], correct.
- X-Intercept: [tex]\( 3(x-2) = 0 \)[/tex] gives [tex]\( x=2 \)[/tex], correct.
This function meets all the required criteria.
### Conclusion
The function that satisfies a horizontal asymptote at [tex]\( y=3 \)[/tex], a vertical asymptote at [tex]\( x=1 \)[/tex], and an [tex]\( x \)[/tex]-intercept at [tex]\( x=2 \)[/tex] is:
[tex]\[ f(x) = \frac{3(x-2)}{x-1} \][/tex]
So, the correct answer is:
[tex]\[ \boxed{4} \][/tex]
