Counter Examples
Mathematics Dictionary
Dr. K. G. Shih
Contents of Mathematics Dictionary
Program WebABC Applications of Mathematics Dictionary on Internet
Graphic Samples of Mathematics Dictinary
Subject : Intersection of Function with its Inverse
Q1. What is the inverse of a function
A1. The defintion is
    * It is the reflection of Y = F(X) about Y = X.
    * It is the image of Y = F(X) about Y = X
    * Function y = x looks like an mirror
Q2. Expression of inverse of y = F(x) = a*x^2+b*x+c
A2. Inverse of Y = a*x^2+b*x+c is X = a*y^2+b*y+c
    * Solve for x and we have x = (-b + Sqr(b^2-4*a*(c-y))/(2*a)
    * Interchange x and y we have
    * The inverse is y = (-b + Sqr(b^2-4*a*(c-x))/(2*a)
    * The inverse is y = (-b - Sqr(b^2-4*a*(c-x))/(2*a)
    * Express x in terms of y
    * The inverse is x = a*y^2 + b*y + c
Q3. Proof inverse is x=a*y^2 + b*y + c
A3. Proof
    * Since y = (-b + Sqr(b^2-4*a*(c-x))/(2*a)
    * Hence 2*a*y + b = Sqr(b^2 - 4*a*(c-x))
    * (2*a*y + b)^2 = b^2 - 4*a*(c-x)
    * 4*a^2*y^2 + 4*a*b*y + b^2) = b^2 - 4*a*c + 4*a*x
    * 4*a*x = 4*a^2*y^2 +4*a*b*y + 4*a*c
    * x = a*y^2 + b*y + c
Q4. Properties of inverse function
A4. Let y=F(x) and inverse y=G(x)
    * F(G(x)) = x
    * G(F(x)) = x
Q5. Example : Prove F(G(x))=x
A5. Proof
    * F(x) = a*x^2+b*x+c and G(x) = (-b+sqr(b^2-4*a*(c-x)))/(2*a)
    * F(G(x))=a*(G(x))^2+ b*G(x)+c
    *
Q6. How many intersctions of y = a*x^2+b*x+c with its inverse

A6. Graphic solutions
Program ABH
    * ABH program 06 01 : There is no intersection
    * ABH program 06 02 : There is one intersection
    * ABH program 06 03 : There are two intersections
    * ABH program 06 04 : There are four intersections
Q7. How to find intersections ?
A7. Find intersection of y = a*x^2 + b*x + c with inverse
    * If function has intersection with its inverse
    * It must intersect with y=x
    * Hence we solve y = a*x^2 + b*x + c and y = x first
    * This gives a quadratic equation : a*x^2 + (b-1)*x + c = 0
    * Let Discriminant D = (b-1)^2 - 4*a*c
      * D < 0 then there is no intersection
      * D = 0 there is one intersection, i.e. y = x is tangent to curve
      * D > 0 there are 2 or 4 intersections
Q8. How to find the 3rd and 4th intersections ?
A8. Solve a quartic equation
    * If D > 0 then we have to solve a quartic equation
    * Substitute y = a*x^2 + b*x + c into x = a*y^2 + b*y + c
    * x = a*(a*x^2+b*x+c)^2 + b*(a*x+b*y+c) + c = 0
    * Expand above equation we get a quartic equation
Q9. How to solve the quatic equation ?
A9. There are 2 methods
    * How to solve the quartic equation ?
    * We alreay have two solutions for this quartic equation
    * They are the solutions of y = a*x^2 + b*x + c and y = x
    * Let the two sultion be r and s
    * Then we have two methods to solve the quartic equation
      * One method is using synthetic division
      * Other one is using the relation of roots and coefficients
Examples
Example 1 : Use Program ABH to view the diagrams
    * y = x^2 + 1 with its inverse ....... 06 01
    * y = x^2 + 0.25 with its inverse .... 06 02
    * y = x^2 with its inverse ........... 06 03
    * y = x^2 - 1 with its inverse ....... 06 04
      * Program 06 01 han no intersection. Why ?
      * Since x = x^2 + 1 or x^2 + x + 1 = 0 has no real roots

      * Program 06 02 han one intersection. Why ?
      * Since x = x^2 + 1/4 or x^2 + x + 1/4 = 0
      * (x + 1/2)^2 = 0 has a real root x = 1/2 and y = 1/2
      * Intersection at (1/2, 1/2)

      * Progra 06 03 han two intersections. Why ?
      * x = x^2 or x^2 + x = 0
      * x*(x+1) = 0 has two real roots x = 0 or x = -1
      * Intersections at (0, 0) and (-1, -1)
      * substitute y=x^2 into x=y^2
      * x = (x^2)^2 or x^4 - x = 0
      * x*(x^3 -1) = 0. It is known (x^3 -1 ) = 0 has only one real
      * Hence it has only two intesections

      * Program 06 04 han four intersections. Why ?
      * x = x^2 - 2 or x^2 + x - 2 = 0
      * (x-1)*(x+2) = 0 has two real roots x = -2 or x = 1
      * Intersections at (-2, -2) and (1, 1)
      * substitute y = x^2 - 2 into x = y^2 - 2
      * x = (x^2-2)^2 - 2 or x^4 - 4*x^2 - x + 2 = 0
      * (x + 1)*(x - 2)*(x^2 + x - 1) = 0
      * x^2 + x - 1 = 0 and x = (-1+Sqr(5))/2 or x = (-1-Sqr(5))/2
      * Hence it has four intesections

Example 2 Sample form program 11 07
Example 3 Solve x^4 - 4*x^2 - x + 2 = 0
    * Sythetic division

    * ... 1 ... 0 ... -4 ... -1 ... 2 ... | ... 2
    * ......... 2 .... 4 .... 0 .. -2
    * -------------------------------------
    * ... 1 ... 2 .... 0 ... -1 ... 0 ... | .. -1
    * ........ -1 ... -1 .... 1
    * -------------------------
    * ... 1 ... 1 ... -1 .... 0

    * Hence (x + 1)*(x - 2)*(x^2 + x - 1) = 0
    * The roots are -1, 2, (-1+sqr(5))/2, (-1-sqr(5))/2

    * Other method : Quartic formula in MD2002
    * Use MD2002 program 17 04
Example 4 x^4 - 4*x^2 - x + 2 = 0 has roots -1 and 2. Find others
    * Use relation of roots and coefficients
    * Sum of roots = p + q + r + s = -b/a
    * Product of roots = p*q*r*s = +e/a
    * Let p=-1 and q=2 then
    * -1 + 2 + r + s = -0/1 or r + s = -1
    * -1*2*r*s = 2/1 or r*s = -1
    * Hence we can find r and s
Example 5 Intersections of y=a*x^2+b*x+c and x=a*y^2+b*y+c
    * x = a*(a*X^2+b*x+c)^2 + b*(a*x^2+b*x+c) + c
    * x = a*(a^2*x^4+b^2*x^2+c^2+2*a*b*x^3+2*a*x^2*c+2*b*x*c)
    * ....+ b*a*x^2+b^2*x+b*c +c
    * a^2*x^4 + 2*a*b*x^3 + (b2+2*a*c+a*b)*x^2 + (b^2+2*b*c-1)*x
    * ....+ (c^2+b*c+c) = 0

    * Coefficients of quartic equation :
    * A = a^2
    * B = 2*a*b
    * C = b^2 + a*b + 2*a*c
    * D = b^2 + 2*b*c -1
    * E = c^2 + b*c + c
Example 6 How many intersections of y=x^2-6*x+8 with its inverse ?
Answer :
  • Use program ABH 06 10 to find graphic solution
  • Use mathematical method to find the solutions
Remarks
    * The author spent more than 10 years to compile Math Dictionary
    * The system is easy in use and gives thousands applications
    * On this website only some samples are given for demonstration
    * You should have it on your computer
    * Then you can use your computer as a graphic calculator

    * Also you can find many mathematical patterns for school wook
    * For more information, Please contact Dr. K. G. Shih
    * Visit the URL to see more samples
    * Visit the URL to view 105 graphic samples in Program WebABC
    * You can download it and Click file WebABC will run
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