We already know that some operations "undo" other operations. We call these inverse operations. Match the operations below with their inverse operation:
| Draggable item | arrow_right_alt | Corresponding Item |
|---|---|---|
subtract | arrow_right_alt | subtract |
add | arrow_right_alt | divide |
square | arrow_right_alt | add |
square root | arrow_right_alt | multiply |
divide | arrow_right_alt | square root |
multiply | arrow_right_alt | square |
For the next two investigations, we will take this a step further. Instead of looking at single inverse operations, we will examine inverse functions. Here is an example of a pair of inverse functions:
We will determine what makes these inverse functions and how we can identify inverse functions.
Let's look at some functions. We'll start easy.
If f(x) = x+4, evaluate f(5)=
Now let's say we have the function g(x) = x-4, what is g(9)?
If f(x)=5x, then what is f(3)?
Now given function g(x) =
Given f(x) = 5x+2, evaluate f(3).
If g(x) =
Look back at questions 2-7.
What do you notice about the relationship between f(x), and the g(x) function that followed?
Q#2-3 Q#4-5 Q#6-7
f(x) = x+4 f(x) = 5x f(x) = 5x+2
f(5) = 9 f(3) = 15 f(3) = 17
g(x) = x-4 g(x) =
g(9) = 5 g(15) = 3 g(17) = 3
You may have noticed that the output of g(x) was the input of f(x), and the output of f(x) was the input of g(x).
When this relationship happens, it means that f(x) and g(x) are inverse functions.
For example : If f(x) was a machine that you put yarn into, and a sweater came out, then the inverse function would be a machine that you put a sweater into, and then yarn comes out.
Instead of using g(x) as the inverse function of f(x), we use the notation f -1(x). We say "f inverse of x". The inverse function of f(x) is f -1(x).
If f(7) = 11, then f -1(11)= ____
With functions that contain only one operation, we can predict what the inverse function will be.
If f(x) = x-6, then what is f -1(x)?
If f(x) =
With more than one operation it can be a bit trickier.
If f(x) = 3x-7, then what is f -1(x)?
If you could make your own non-math inverse machines what would they do?