To determine the asymptotes of the function \( F(x) = \frac{3x - 1}{x - 1} \), we need to identify both the vertical and horizontal (or oblique) asymptotes.
1. Vertical Asymptotes
Vertical asymptotes occur where the function is undefined, which typically happens when the denominator equals zero. Here, we set the denominator equal to zero:
\[ x - 1 = 0 \implies x = 1 \]
So, there is a vertical asymptote at \( x = 1 \).
2. Horizontal or Oblique Asymptotes
Next, we examine the behavior of the function as \( x \) approaches infinity to determine horizontal or oblique asymptotes.
To find the horizontal asymptote, we compare the degrees of the polynomial in the numerator and the degree of the polynomial in the denominator:
- The numerator \( 3x - 1 \) has a degree of 1.
- The denominator \( x - 1 \) also has a degree of 1.
Because the degrees of the numerator and the denominator are the same, we can find the horizontal asymptote by taking the ratio of the leading coefficients:
\[ y = \frac{\text{Leading coefficient of numerator}}{\text{Leading coefficient of denominator}} = \frac{3}{1} = 3 \]
Thus, there is a horizontal asymptote at \( y = 3 \).
Summary
The function \( F(x) = \frac{3x - 1}{x - 1} \) has:
- A vertical asymptote at \( x = 1 \).
- A horizontal asymptote at \( y = 3 \).