Expert: Paul Klarreich - 12/7/2007

Question
Dear Mr. Klarreich,
I would deeply appreciate if you could help me with the following two calculus AB problems:
1. Let f be the function defined by f(x)=-2+ ln(x^2)
a) For what real numbers x is f defined?
b) Find the zeros of f.

2. Let f be the function given by f(x)= ((9x^2)-36)/ ((x^2)-9)
a) Describe the symmetry of the graph of f.
b. Write an equation for each vertical and each horizontal asymptote of f.
c. Find the intervals on which f is increasing.

Thank you so much for your help!
Sincerely,
Patsy

Answer
Questioner:   Patsy
Category:  Advanced Math
Private:  No
 
Subject:  help with calculus AB problems
Question:  Dear Mr. Klarreich,
I would deeply appreciate if you could help me with the following two calculus AB problems:
1. Let f be the function defined by f(x)=-2+ ln(x^2)
a) For what real numbers x is f defined?
b) Find the zeros of f.

2. Let f be the function given by f(x)= ((9x^2)-36)/ ((x^2)-9)
a) Describe the symmetry of the graph of f.
b. Write an equation for each vertical and each horizontal asymptote of f.
c. Find the intervals on which f is increasing.

Thank you so much for your help!
Sincerely,
Patsy
 
........................................
1. Let f be the function defined by f(x)=-2+ ln(x^2)
a) For what real numbers x is f defined?
b) Find the zeros of f.

a) Better you should ask:"For what real numbers x is f NOT defined?"  Once you answer this, you have your answer:

ln(...) is defined for all positive arguments.  (not zero, not negative.)

x^2 is never negative, but could be zero.  So your answer is:

f(x) is defined for all x /= 0.

b) The zeroes of f are the solutions to the equation  f(x) = 0.

-2 + ln(x^2) = 0

Use ln properties:

ln(x^2) = 2 ln |x|

-2 + 2 ln |x| = 0

2 ln |x| = 2

ln |x| = 1

|x| = e

x = +- e.

Those are your zeroes.
................................

2. Let f be the function given by f(x)= ((9x^2)-36)/ ((x^2)-9)

>> You were careful about parentheses.  Very good.

a) Describe the symmetry of the graph of f.
b. Write an equation for each vertical and each horizontal asymptote of f.
c. Find the intervals on which f is increasing.
        9x^2 - 36
f(x) = -------------
        x^2 - 9


Since f(-x) clearly = f(x), you have left-right symmetry.  [You only have to analyze positive x-values.]

b. Vertical:  

       9(x + 2)(x - 2)
f(x) = ----------------
        (x+ 3)(x - 3)

This 'blows up' at  x = +- 3. Those are your vertical asymptotes.
To find H.A.:

Rewrite f(x): [Divide every term by x^2.]


       9 - 36/x^2
f(x) = -------------
        1 - 9/x^2
           9 - 0
Lim f(x) = ------- = 9
x-> inf     1 - 0

y = 9 is your H.A.

c.  find f'(x)

       (x^2 - 9)(18x) - (9x^2 - 36)(2x)
f'(x) = ---------------------------------
                  ()^2


       2x(9x^2 - 81 - 9x^2 + 36)
f'(x) = -------------------------
                  ()^2

       2x(- 45)
f'(x) = ----------
          ()^2

Now if x > 0, that is negative.  So the graph is falling for x >0, rising for x < 0.

There are some free graphing programs on the web.  Try gcalc.net.  Use those for a quick-and-dirty check on your work.