Consider the function f(x)= abs(x)(x-3)/9-x^2
a) what is the domain of f? What are the zeros of f?
b) find all nonremovable discontinuities of f
c) determine all vertical and horizontal asymptotes of f
f(x) = | (x)(x-3) / (9 - x^2) |
(a) Domain & Zeroes of f?
Domain is the set of all possible values of x. In the given function, note that the denominator can not be equal to zero, because any number divided by zero results to indeterminate (if 0/0) or infinity (if any non-zero number/0). Thus we get the values of x in which the denominator will be zero and restrict the domain on those values:
(9 - x^2) = 0
(3 - x)(3 + x) = 0
x = 3 and -3
Therefore, domain is all real numbers except 3 and -3.
Zero of a function is all x values that results to y = 0. Thus, we just substitute f(x) = 0 and solve for x:
0 = | (x)(x-3) / (9 - x^2) |
0 = (x)(x-3) / (x-3)(x+3)(-1)
0 = x / (-1)(x+3)
x = 0
Thus, f(x) = 0 at x = 0.
(b) Nonremoval discontinuities of f?
Note that we can further simplify the given function:
f(x) = | (x)(x-3) / (9 - x^2) |
f(x) = | (x)(x-3) / (x-3)(x+3)(-1) |
f(x) = | x / (-1)(x+3) |
Nonremovable discontinuity therefore occurs at x = -3 (the denominator becomes zero).
Note that x = 3 is a REMOVABLE discontinuity, because there is a factor at the numerator that is cancelled in the denominator in which when the x value is substituted, the result is 0/0.
(c) Asymptotes of f?
i. Vertical asymptote
We already got this one. Vertical asymptote occurs at x = -3 (the nonremovable discontinuity). At this point the f(x) approaches infinity.
ii. Horizontal asymptote
We can do the limits here. We find the limit of f(x) as x -> infinity:
lim | (x)(x-3) / (9 - x^2) | as x->infinity
lim | x / (-1)(x+3) |
lim | -x / (x+3) |
What we can do here is use L'hopital's rule or much easier, we can divide the numerator and denominator by the x with the highest degree (which is x):
lim | (-x/x) / (x+3)/x |
lim | -1 / (1 + 3/x) |
3/x as x -> is zero, leaving
| -1/1 | or 1
Thus at x -> infinity, f(x) = 1
Long solution! OwO
Hope this helps~ ;)