SOLUTION: How do I plot {{{f(x) = 4x - 3/x}}}? Why doesn't the y-intercept 0,0 work in this case?

Algebra ->  Coordinate-system -> SOLUTION: How do I plot {{{f(x) = 4x - 3/x}}}? Why doesn't the y-intercept 0,0 work in this case?      Log On


   

Question 960716: How do I plot f%28x%29+=+4x+-+3%2Fx? Why doesn't the y-intercept 0,0 work in this case?
Answer by josgarithmetic(39621) About Me  (Show Source):
You can put this solution on YOUR website!
f is undefined at x=0, because denominators must not become zero.

You could make x y table of values, but you could also condense the function expression and evaluate a few things about it.

First notice that as x goes unbounded to the left or the right, f(x) will approach 4x, so this will be a slant asymptote, of y=4x.

4x-3%2Fx
4x%28x%2Fx%29-3%2Fx
highlight%28%284x%5E2-3%29%2Fx%29

Where is the numerator 0?
4x%5E2=3
x%5E2=3%2F4
x=3%2F2 or x=-3%2F2, so these are critical points, as is x=0.

The three critical x values form intervals:
-infin%3Cx%3C-3%2F2
-3%2F2%3Cx%3C0
0%3Cx%3C3%2F2
3%2F2%3Cx%3Cinfin
Evaluate the sign of f in each interval. That will help to determine if f is above or below the horizontal axis.


graph%28300%2C300%2C-10%2C10%2C-10%2C10%2C4x-3%2Fx%29

A vertical asymptote in addition to the slant asymptote, is x=0.