f(x)= 3x+9/(x^2-100)
If this function has a horizontal asymptote, give its y-value
Find the x-intercept(s).
Find the y-intercept
To find the horizontal asymptote of the given function f(x) = (3x + 9) / (x^2 - 100), we need to analyze the function's behavior as x approaches positive or negative infinity.
1. Analyzing the behavior as x approaches infinity:
As x approaches positive or negative infinity, the term with the highest degree in the denominator (x^2) becomes dominant. Therefore, we can ignore the other terms and simplify the function to:
f(x) ≈ (3x) / (x^2)
≈ 3 / x
As x approaches infinity, the function f(x) approaches 0, so the horizontal asymptote is y = 0.
2. Finding the x-intercept(s):
To find the x-intercept(s), we set f(x) equal to zero and solve for x:
3x + 9 / (x^2 - 100) = 0
Since the numerator of the fraction (3x + 9) cannot be zero, we focus on the denominator and solve for x:
x^2 - 100 = 0
x^2 = 100
x = ±10
The x-intercepts are x = 10 and x = -10.
3. Finding the y-intercept:
To find the y-intercept, we set x = 0 in the function f(x):
f(0) = (3(0) + 9) / ((0)^2 - 100)
= 9 / (-100)
= -0.09
The y-intercept is y = -0.09.