badaxeff said:
a. Find the rational function f(x) = P(x)/Q(x) which satisfies the following conditions:
1. f(x) has a vertical asymptotes at x = + & - 2
2. f(x) has the oblique asymptote y=2(x+3)
3. f(x) passes throught the points (4, 105/8) and (-4, 33/8)
4. f(x) has the domain {x | x is real and x cant be + or - 2
5. f(x) contains no irricucible quadriatics
6. all coefficients are real
b. find all roots of f(x)
I just need some help, I totally dont understand! Please help!
That is quite a few conditions. Put them together and see what you get. You must understand the implications of each.
f(x) = P(x)/Q(x) -- It's a Rational Function. This make both P(x) and Q(x) polynomials. So far, we do not know the degree of these polynomials.
1. f(x) has a vertical asymptotes at x = + & - 2
The Q(x) contains (x+2) and (x-2) - It's degree is AT LEAST 2
2. f(x) has the oblique asymptote y=2(x+3)
An oblique, linear asymptote implies that the degree of P(x) is one (1) greater than the degree of Q(X). No we know the degree of P(x) is AT LEAST 3. This also means that f(x) can be written as f(x) = 2(x+3) + Constant/Q(x). There is more information here, but let's get back to it.
3. f(x) passes throught the points (4, 105/8) and (-4, 33/8)
Obviously, this means f(4) = 105/8 and f(-4) = 33/8, but there is more here than that. Looking at the assymptote, we see that 2(4+3) = 14 > 105/8, so the graph is above the asymptote as x increases in the positive direction. Also, 2(-4+3) = -2 < 33/8, so f(x) is below the asymptote as x increases in the negative direction. This may be more important than it seems, at first. With two assymptotes, linear factors in the denominator might not be able to do this. We'll have to keep an eye on it.
4. f(x) has the domain {x | x is real and x cant be + or - 2
Not a lot more information, here. We sort of already knew that.
5. f(x) contains no irricucible quadratics
Good. Everything can be factored - Maybe. It does not promise irreducible polynomials of degree greater than 2. It may have meant that, but that isn't what it says.
6. all coefficients are real
Good. Everything that can be factored will require only Real Numbers.
Most of this exercise is seeing the implications of the given information. See where that leads you.
