Question 1173303: Consider the two functions: f(x)=x^2-4 and g(x)=2x^3-3x^2-5x+6
A. A function is defined by h(x)=f(x)/g(x). Determine the domain of this function.
B. What are the discontinuities? (remember the discontinuities includes the removable ones and the asymptotes).
C. State the asymptotic behaviour
D. Find all intercepts.
E. Sketch h(x)
Answer by greenestamps(13200)   (Show Source):
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The critical points on the graph are at any value of x that makes a factor of the numerator or denominator equal to 0: -2, -3/2, 1, and 2.
A. Domain: any value of x except those that make the denominator 0. So all x except 1, 2, and -3/2.
B. Discontinuities: wherever there is a value of x that is excluded from the domain: at x=1, x=2, and x=-3/2.
The discontinuity at x=2 is removable (there is a hole in the graph); the discontinuities at x=1 and x=-3/2 are asymptotes.
C. Asymptotic behavior: we'll come back to that after part D.
D. The y-intercept is at (0,f(0)) = (0,-4/6) = (0,-2/3).
The only x-intercept is where the numerator is 0 and the denominator is not also 0 -- at (-2,0).
Asymptotic behavior: The function value is positive for all x greater than 2 (the largest critical value). Since the only x-intercept is at x=-2, the behavior is:
towards +infinity as x approaches 2 from the right
towards -infinity as x approaches 2 from the left
towards -infinity as x approaches -3/2 from the right
towards +infinity as x approaches -3/2 from the left
Note the graph is asymptotic to 0 from above as x approaches +infinity; it is asymptotic to 0 from below as x approaches -infinity
E. Graph:
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