function help please

Question

Let f(x)=   6x^2-6      

              x^2-7x+10

(a) Write f(x) with the numerator and denominator completely factored.

(b) State the domain.

(c) State the vertical asymptote(s).

(d) State the horizontal asymptote.

(e) State the y-intercept.

(f) State the x-intercept(s).

Answer 1

a. 6*(x-1)(x+1)

      ^{--------------}

       (x-2)(x-5)

 

b. x not equal to 2 or 5, x∈R

 

c. at x=2 and x=5

 

d. at x→∞, f(x)→6

 

e.x=0 f(0)=-3/5

 

f.f(x)=0 at x=±1

 

Jim

 

Answer 2

f(x) = (6x²-6)/(x²-7+10)

 

a) Write f(x) with the numerator and denominator completely factored

(6x2-6) factors to 6(x²-1) = 6(x+1)(x-1)

x²-7+10 factors to (x-5)(x-2)

 

f(x) = 6(x+1)(x-1)/(x-5)(x-2)

 

You might want to check out this tutorial on factoring to learn how to do it for yourself.

 

http://www.wyzant.com/resources/lessons/math/algebra/factoring

 

(b) State the domain

Division by zero is undefined, so the denominator cannot = 0.  It equals zero when x=5 or x=2.  Hence the domain is all real numbers except 2 and 5.

 

(-∞,2)U(2,5)U(5,+∞)           [interval notation] 

 

(c) State the vertical asymptote(s)

Vertical asymptotes exist where the denominator goes to 0.  Hence we have one at x=2 and another at x=5.

 

(d) State the horizontal asymptote.

Horizontal asymptotes occur when x approaches ±∞.  As x gets very large or small (x+1), (x-1), (x-5), and (x-2) all approach x.  Picture x as 10.000.000.  The difference between 99,999,995 (x-5) and 10,000,001 (x+1) is very small.  Hence the x terms in the numerator and denominator cancel leaving only the 6. So there is a horizontal asymptote at y=6.

 

(e) State the y-intercept.

The y-intercept occurs when x=0.  Plug zero in for x in the factored equation for f(x) in step (a) to compute the y-intercept.

 

(f)  State the x-intercept(s)

The x-intercepts occur when y=0.  This can only happen when the numerator of f(x) = 0.  What values of x will cause the numerator of f(x) to be zero?  There are two answers.  Use the factored form of f(x) in (a).