Positive Functions

The graph of a function is all we need to determine where it is positive.

Find the intervals on the x axis over which the polynomial funtion f(x) = x³ + 2x² - 3x - 6 is postive.

• Graph the function (see Graphing Functions)
  
  
  
• The parts of the graph above highlighted in blue are the parts of the graph that are postive (y coordinates are positive). The intervals on the x-axis where the function is positive is highlighted in red. These intervals are the ones we want to find.
  
  
• The red interval on the left goes from one negative root to the next negative root (see Zoom on right). We find these to be x=-2 and -1.73205080757 (see Finding Roots of a Functin). The red interval on the right goes from the positive root to plus infinity. We find this to be x=1.73205080757 (see Finding Roots of a Function).
  

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Over what intervals is the rational funtion f(x) = positive?

• Graph the function;
  
  

• The part of the graph above highlighted in blue is the part of the graph that is positive (y coordinates is positive). The interval on the x-axis where the function is positive is highlighted in red. These are the intervals we want to find.
  
  
• The red interval on the left goes from negative infinity to the negative root and the one on the right goes from the positive root to plus infinity. All we need to do is find the roots (see Finding Roots of a Function). The function is positive of the intervals (-inf, -3/2) and (3/2, inf).