f(x)=(6x^3–9x^2–3x–1)/(4x^2+7x–3)
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Find the equation of the non-vertical asymptote.
y = ____3x/2 - 39/8_____
Does f(x) intersect its non-vertical asymptote? (yes or no) ___YES____
What is the smallest value of x at which f(x) intersects its non-vertical asymptote? ____?____???
I did almost the same question here
http://www.jiskha.com/display.cgi?id=1311223818
follow the procedure shown in this problem
(I had y = (3/2)x + 9/2 as the non-vertical asymptote
The long division is messy with awful fractions)
Since the denominator cannot be zero, the function cannot have any vertical asymptotes, so it is continuous. Thus the graph will stay on one side of the asymptote and never cross y = (3/2)x + 9/2
I understood the first two questions and got them right. But can't get the last.... "What is the smallest value of x at which f(x) intersects its non-vertical asymptote? ___?"
To find the equation of the non-vertical asymptote of the function f(x), we need to determine the limit as x approaches positive or negative infinity.
Step 1: Factorize the numerator and denominator.
The numerator can be written as (6x^3 – 9x^2 – 3x – 1), and the denominator can be written as (4x^2 + 7x – 3).
Step 2: Cancel out common factors.
If there are any common factors in the numerator and denominator, you can cancel them out. However, in this case, there are no common factors.
Step 3: Determine the degree difference.
Compare the degrees of the numerator and the denominator. In this case, the degree of the numerator is 3, and the degree of the denominator is 2. The difference is 3 – 2 = 1.
Step 4: Find the non-vertical asymptote.
The non-vertical asymptote will be a line of the form y = mx + b, where m is the coefficient of the leading term of the numerator divided by the leading term of the denominator.
In this case, the leading term of the numerator is 6x^3, and the leading term of the denominator is 4x^2.
So, the coefficient of the leading term of the numerator divided by the leading term of the denominator is 6/4 = 3/2.
The non-vertical asymptote equation is therefore y = (3/2)x + b.
To find the value of b, we need to determine the y-intercept.
Step 5: Find the y-intercept.
To find the y-intercept, substitute x=0 into the original function and solve for y.
f(0) = (6(0)^3 – 9(0)^2 – 3(0) – 1) / (4(0)^2 + 7(0) – 3)
f(0) = -1 / -3
f(0) = 1/3
So, the y-intercept is 1/3.
Therefore, the non-vertical asymptote equation is y = (3/2)x + (1/3).
Next, let's determine if the function intersects its non-vertical asymptote.
To do this, we can compare the graph of the function f(x) with its non-vertical asymptote.
If the function intersects the non-vertical asymptote, there will be values of x for which f(x) is equal to the equation of the non-vertical asymptote.
To find these values of x, we need to solve the equation:
(6x^3 – 9x^2 – 3x – 1) / (4x^2 + 7x – 3) = (3/2)x + (1/3)
Unfortunately, this equation does not have a clear and simple solution. To find the smallest value of x at which the function intersects with its non-vertical asymptote, you would need to solve this equation numerically using methods like numerical approximation or graphing technology.
So, the answer to the smallest value of x at which f(x) intersects its non-vertical asymptote is unknown without further calculations.