How is a sign chart used to determine intervals where a function is positive and negative if the function has no zeros?
For example, f(x)=-6/(2x-3). Obviously the function has no zeros so I'm confused as to how I would use a sign chart. Please help!
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You are asked to find the intervals where a function is positive or negative, but the function has no zeros (x-intercepts).
(1) If the function is continuous (naively it can be drawn without lifting your pencil) then the function is either always positive or always negative. For example `y=sqrt(x)+1` is always positive while `y=-x^2-1` is always negative. Also a function like `y=1/x^2+1` is always positive.
(2) If the function is not continuous, then you have to check the intervals defined by the discontinuities. Things that can cause discontinuities are division by zero or taking the logarithm of zero. Also, piecewise functions might have jump discontinuities built in.
Ex: `y=-x^2-1` for `x<=0` , `y=x^2+1` for x>0 has this graph:
In your example `y=(-6)/(2x-3)` the function is never zero. But there is a discontinuity, a break in the graph, at `x=3/2` since you cannot divide by zero. So we must check the sign on either side of `x=3/2` . We can choose any point on the intervals, so at `x=0,f(0)=2>0` so the function is positive for `x<3/2` or on the interval `(-oo,3/2)` . For `x=2,f(2)=-6<0` so the function is negative for `x>3/2` or on the interval `(3/2,oo)` . (Note that the function cannot change sign again -- there are no more breaks in the graph, and it doesn't cross zero.)
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