f(x)=x^3-3x^2-3x-8/(-3x^2-4x-6)
Find the equation of the non-vertical asymptote.
What is the smallest value of x at which f(x) intersects its non-vertical asymptote?
the non vertical asymptote is -1/3x + 13/9 i found that using synthetic division and i know there is an intersect its non-vertical asymptote but i don't know how to find it....
I agree on the asymptote.
So set the intercepts equal..
-x/3+ 13/9= (x^3-3x^2-3x-8)/(-3x^2-4x-6)
+x^3 +4x^2/3+2x-13/3 x^2 -52/9x-26/3=x^3-3x^2-3x-8
combining terms
x^2 (4/3-13/3+3) +x(2+3 -52/9)-26/3+8=0
check that several times. then use the quadratic equation to solve for x
To find the intersection of the function f(x) with its non-vertical asymptote, we can set the function equal to the equation of the asymptote and solve for x.
In this case, the equation of the non-vertical asymptote is given as -1/3x + 13/9.
So, we can set f(x) equal to -1/3x + 13/9 and solve for x.
To do that, we substitute f(x) = -1/3x + 13/9 into the equation of f(x):
x^3 - 3x^2 - 3x - 8 / (-3x^2 - 4x - 6) = -1/3x + 13/9
Next, we simplify the expression by multiplying through by the denominator (-3x^2 - 4x - 6):
(x^3 - 3x^2 - 3x - 8) = (-1/3x + 13/9) * (-3x^2 - 4x - 6)
After multiplying and simplifying, we should have a cubic equation in terms of x. To solve for x, we can rearrange the equation to have all terms on one side:
0 = x^3 - 3x^2 - 3x - 8 + (1/3)(x^3) - (13/9)(x^2) - (13/9)(x)
Combining like terms:
0 = (4/3)x^3 - (29/9)x^2 - (40/9)x - (35/9)
At this point, the equation needs to be solved numerically. One common method is to use a graphing calculator or a computer algebra system to find the real roots. Alternatively, you can use methods such as Newton's method or the bisection method to approximate the roots.
Once you find the values of x where the function intersects its non-vertical asymptote, you can evaluate the function at those x-values to find the corresponding y-values.