Show all work to identify the asymptotes and zero of the function f(x) = 3x over x^2 - 9
Plz help
zero when the numerator is zero: x = 0
vertical asymptotes when the denominator is zero: x = ±3
horizontal asymptote is y=0 because the degree of the numerator is less than that of the denominator.
To identify the asymptotes and zeros of the function f(x) = 3x / (x^2 - 9), we first look at the denominator (x^2 - 9) to determine if there are any vertical asymptotes.
Vertical Asymptotes:
The denominator (x^2 - 9) can be factored as (x - 3)(x + 3). Vertical asymptotes occur where the denominator equals zero. So, x - 3 = 0 or x + 3 = 0.
Solving x - 3 = 0 gives x = 3.
Solving x + 3 = 0 gives x = -3.
So, the vertical asymptotes are x = 3 and x = -3.
Zeros:
To find the zeros of the function, we set the numerator (3x) equal to zero.
3x = 0
x = 0
So, the zero of the function is x = 0.
Summary:
Vertical Asymptotes: x = 3 and x = -3
Zero: x = 0
To find the asymptotes and zeros of the function f(x) = 3x / (x^2 - 9), we will follow these steps:
1. Determine the vertical asymptotes:
- Vertical asymptotes occur when the denominator of the function equals zero, which in this case is (x^2 - 9).
- So, we must solve the equation x^2 - 9 = 0 for x.
- Factoring the equation, we get (x - 3)(x + 3) = 0.
- Setting each factor equal to zero, we get x - 3 = 0 and x + 3 = 0.
- Solving these equations, we find x = 3 and x = -3, respectively.
- Therefore, we have two vertical asymptotes at x = 3 and x = -3.
2. Determine the horizontal asymptote:
- Horizontal asymptotes occur when the degree of the numerator and the denominator are the same.
- In this case, the degree of the numerator is 1 (x^1) and the degree of the denominator is 2 (x^2).
- Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at y = 0.
3. Determine the zeros of the function:
- Zeros occur when the numerator of the function equals zero, which in this case is 3x.
- Setting 3x equal to zero, we get 3x = 0.
- Solving for x, we find x = 0.
- Therefore, the zero of the function is at x = 0.
To summarize:
- Vertical asymptotes: x = 3 and x = -3
- Horizontal asymptote: y = 0
- Zero: x = 0