W2. Let f(x) =
x
3+9
x
x2
−
3x−
4
,then
(a) find the domain of f and its zero(s).
(b) find the intercept(s)of f if exists.
(c) find the asymptote(s) of f if exists.
(d) sketch the graph of f.
(a) To find the domain of f, we need to look at the values that x cannot take. Since the denominator of the fraction cannot be zero, we need to find the values of x that make the denominator zero:
x^2 - 3x - 4 = 0
(x-4)(x+1) = 0
x = 4 or x = -1
So the domain of f is all real numbers except x = 4 and x = -1. The zeros of f are the values of x that make the numerator of the fraction zero:
x^3 + 9x = x(x^2 + 9) = 0
x = 0
Therefore, the zero of f is x = 0.
(b) To find the x-intercepts, we set f(x) = 0 and solve for x:
x^3 + 9 = x(x^2 - 3x - 4) = 0
x = 0 (this intercept was found in part (a))
(c) To find the asymptotes of f, we need to examine the behavior of the function as x approaches infinity or negative infinity.
As x approaches infinity, the function will approach the value of x^3/x^2 = x. So, there is a horizontal asymptote at y = x.
There are no vertical asymptotes since we found the zeros of the denominator in part (a).
(d) To sketch the graph of f, it would look like a curve that approaches the asymptote y=x as x approaches infinity or negative infinity. The function will also have a zero at x=0. The graph will not cross the x-axis at x=0, as it has a zero but not an x-intercept.