In What is an inverse function? I proposed a way of teaching this concept starting with its graphical representations using GeoGebra applets. Al-Zboun Lilliana in our Linkedin group shares her idea for introducing the inverse function. She says that the most difficult part in teaching this concept is to make it make sense to students and not so much in making the students understand its definition or teaching them the process of finding the inverse function of a given function(by a graph or by a formula) or to “verify algebraically” that the functions are inverses.
Here’s her proposed teaching sequence starting with examples that students can relate to in Levels 1 and 2. I would suggest inserting the activities I described in What is an inverse function? before Level 3 which introduces the algebraic solution.
Examples SET(1)-Level 1:
1. If we need to call someone we are asking for her/his name on the list of our phone contacts …
2. If someone of our contacts is calling us “our phone shows who is calling” This is the job of an inverse function: “finding the name corresponding to the number”
Examples SET(2)- Level 2:
1. If George makes $100/day. We know how to answer questions such as “After 7 days, how much money has he made?” We use the function W(t)=100t
But suppose I want to ask the reverse question:
2. “If George has made $700, how many hours has he worked?” The students know the answer : Time : t(W)=W/100. Given any amount of money, divide it by 100 to find how many days he has worked.
This is the job of an inverse function. It gives the same relationship, but reverses the dependent and independent variables.
Examples SET(3)- Level 3:
In this set the teacher includes examples to show simplifying solutions of mathematical questions
Example (1): Solve log (3 x – 2) = 3
• Since logarithmic and exponential functions are inverses of each other, we can write the following.
a = log (b) if and only b = 10^a
• Use the above property of logarithmic and exponential functions to rewrite the given equation as follows.
3x – 2 = 10^3
• Solve for x to obtain.
3x = 1002
x= 1002÷3=334
Example (2): Find the Range of the function ( or any RATIONAL function) :
F(x)= (3x+1)/(3 -x) or [y=(3x+1)/(3 -x)]
• Since the RANGE of a one to one function is the DOMAIN of its inverse. Let us first show that function f given above is a one to one function.
• Hence the given function is a one to one. let us find its inverse.
• Interchange x and y and solve for y.
x =(3y+1)/(3 -y)
And find y = (3x-1)/(3+x)
The inverse g(x) of function f(x) is given by.
g(x) = (3x-1)/(3+x)
• The domain of g(x) is R except x = -3. Hence THE RANGE of f(x) is R/{-3}.