To determine the vertical and horizontal asymptotes for the function [tex]\( f(x) = \frac{3x^2}{x^2 - 4} \)[/tex], follow these steps:
### Step 1: Finding Vertical Asymptotes
Vertical asymptotes occur where the denominator of the function goes to zero, provided the numerator does not also go to zero at those points.
The function given is:
[tex]\[ f(x) = \frac{3x^2}{x^2 - 4} \][/tex]
We set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ x^2 - 4 = 0 \][/tex]
Solving this equation:
[tex]\[ x^2 - 4 = (x - 2)(x + 2) = 0 \][/tex]
[tex]\[ x - 2 = 0 \quad \text{or} \quad x + 2 = 0 \][/tex]
[tex]\[ x = 2 \quad \text{or} \quad x = -2 \][/tex]
So, the vertical asymptotes are at [tex]\( x = 2 \)[/tex] and [tex]\( x = -2 \)[/tex].
### Step 2: Finding Horizontal Asymptote
To find the horizontal asymptote for the function, we compare the degrees of the numerator and the denominator.
The function given is:
[tex]\[ f(x) = \frac{3x^2}{x^2 - 4} \][/tex]
- The degree of the numerator [tex]\( 3x^2 \)[/tex] is 2.
- The degree of the denominator [tex]\( x^2 - 4 \)[/tex] is also 2.
When the degrees of the numerator and denominator are equal, the horizontal asymptote is determined by the ratio of the leading coefficients of the numerator and the denominator.
The leading coefficient of the numerator [tex]\( 3x^2 \)[/tex] is 3.
The leading coefficient of the denominator [tex]\( x^2 \)[/tex] is 1.
Thus, the horizontal asymptote is:
[tex]\[ y = \frac{3}{1} = 3 \][/tex]
### Conclusion
The vertical asymptotes are [tex]\( x = -2 \)[/tex] and [tex]\( x = 2 \)[/tex], and the horizontal asymptote is [tex]\( y = 3 \)[/tex].
Therefore, the correct option is:
Horizontal asymptote: [tex]\( y = 3 \)[/tex]
Vertical asymptote: [tex]\( x = -2, x = 2 \)[/tex]
So, the correct choice is:
horizontal asymptote: [tex]\( y = 3 \)[/tex] vertical asymptote: [tex]\( x = -2, x = 2 \)[/tex]
