Write Rational Functions  Problems With Solutions
Write rational functions given their characteristics such as vertical asymptotes, horizontal asymptote, x intercepts, hole.
Problem 1:
Write a rational function f that has a vertical asymptote at x = 2, a horizontal asymptote y = 3 and a zero at x =  5.
Solution to Problem 1:

Since f has a vertical is at x = 2, then the denominator of the rational function contains the term (x  2). Function f has the form.
f(x) = g(x) / (x  2)

g(x) which is in the numerator must be of the same degree as the denominator since f has a horizontal asymptote. Also g(x) must contain the term (x + 5) since f has a zero at x =  5. Hence
f(x) = 3 (x + 5) / (x  2)

Check that all the characteristics listed in the problem above are in the graph of f shown below.
.
Problem 2:
Write a rational function g with vertical asymptotes at x = 3 and x = 3, a horizontal asymptote at y = 4 and with no x intercept.
Solution to Problem 2:

Since g has a vertical is at x = 3 and x = 3, then the denominator of the rational function contains the product of (x  3) and (x + 3). Function g has the form.
g(x) = h(x) / [ (x  3)(x + 3) ]

For the horizontal asymptote to exist, the numerator h(x) of g(x) has to be of the same degree as the denominator with a leading coefficient equal to 4. At the same time h(x) has no real zeros. Hence
f(x) = [ 4x^{ 2}  6 ] / [ (x  3)(x + 3) ]

Check the characteristics in the graph of g shown below.
.
Problem 3:
Write a rational function h with a hole at x = 5, a vertical asymptotes at x = 1, a horizontal asymptote at y = 2 and an x intercept at x = 2.
Solution to Problem 3:

Since h has a hole at x = 5, both the numerator and denominator have a zero at x = 5. Also the vertical asymptote at x = 1 means the denominator has a zero at x = 1. An x intercept at x = 2 means the numerator has a zero at x = 2. Finally the horizontal asymptote y = 2 means that the numerator and the denominator have equal degrees and the ratio of their leading coefficients is equal to 2. Hence
h(x) = [ 2 (x  5)(x  2) ] / [ (x  5)(x + 1) ]

The graph of h is shown below, check the characteristics.
.
Problem 4:
Write a rational function f with a slant asymptote y = x + 4, a vertical asymptote at x = 5 and one of the zeros at x = 2.
Solution to Problem 4:

The graph of f has a slant asymptote y = x + 4 and a vertical asymptote at x = 5, hence f(x) may be written as follows
f(x) = (x + 4) + a / (x  5)

where a is a constant to be determined using the fact that f(2) = 0 since f has a zero at x = 2.
f(2) = (2 + 4) + a / (2  5) = 0

Solve the above for a to obtain.
a = 18

Hence f(x) is given by.
f(x) = (x + 4) + 18 / (x  5) = (x^{ 2}  x  2) / (x  5)

Check the characteristics of the graph of f shown below.
.
More math problems with detailed solutions in this site.


Linear ProgrammingNew !
Online Step by Step Calculus Calculators and SolversNew !
Factor Quadratic Expressions  Step by Step CalculatorNew !
Step by Step Calculator to Find Domain of a Function New !
Free Trigonometry Questions with Answers

Interactive HTML5 Math Web Apps for Mobile LearningNew !

Free Online Graph Plotter for All Devices
Home Page 
HTML5 Math Applets for Mobile Learning 
Math Formulas for Mobile Learning 
Algebra Questions  Math Worksheets

Free Compass Math tests Practice
Free Practice for SAT, ACT Math tests

GRE practice

GMAT practice
Precalculus Tutorials 
Precalculus Questions and Problems

Precalculus Applets 
Equations, Systems and Inequalities

Online Calculators 
Graphing 
Trigonometry 
Trigonometry Worsheets

Geometry Tutorials 
Geometry Calculators 
Geometry Worksheets

Calculus Tutorials 
Calculus Questions 
Calculus Worksheets

Applied Math 
Antennas 
Math Software 
Elementary Statistics
High School Math 
Middle School Math 
Primary Math
Math Videos From Analyzemath
Author 
email
Updated: February 2015
Copyright © 2003  2015  All rights reserved