Unit 1

Functions

If the degree of the numerator is equal to the degree of the denominator, such as in «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfrac»«mrow»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»5«/mn»«mi»x«/mi»«mo»-«/mo»«mn»1«/mn»«/mrow»«mrow»«mn»2«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»6«/mn»«mi»x«/mi»«mo»+«/mo»«mn»9«/mn»«/mrow»«/mfrac»«/mstyle»«/math», the numerator and denominator will grow at a similar rate and will approach the ratio of the leading coefficients, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfrac»«mn»3«/mn»«mn»2«/mn»«/mfrac»«/mstyle»«/math» in this case.


Summary
  • Factor a rational function fully to determine the non-permissible values on the domain of the function.
  • Non-permissible values in a rational function result in a break in the graph of the function or a vertical asymptote.
  • If a factor occurs in both the numerator and denominator, the corresponding non-permissible value will result in a point of discontinuity on the graph of the function.
  • The behaviour of a rational function near a point of discontinuity does not include sharp increases or decreases in the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-value. A specific «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-value is approached.
  • If a factor is in the denominator, but not in the numerator, the corresponding non-permissible value will result in a vertical asymptote on the graph of the function.
  • The behaviour of the function near a vertical asymptote includes a sharp increase or decrease in «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-values.
  • Horizontal asymptotes can be predicted by looking at the degree of the numerator and the degree of the denominator.