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Inverse Functions
One-to-One Functions
~ A function whose domain maps onto one and only one element in the range.
~ If we were to redefine the range as the domain and the domain as the range, the elements in the domain (the previous range) are still mapped onto one and only one element in the range ( the previous domain).
Diagram #1 is a function, but NOT 1-to-1.
If we switch the domain with the range, we see that 5 would be mapped onto both 4 and 11, which wouldn't be a function.
Therefore, we wouldn't be able to define its inverse.
Diagram #2 is a function, and it is 1-to-1. If we switch the domain with the range, we would still have a function, therefore, we can declare it as a function.
Horizontal Line Test
To determine if a graph was a function, we used the vertical line test.
The horizontal line test is used to determine if the graph of a function is 1-to-1.
The left graph uses the horizontal line test.
No matter how many horizontal lines are drawn, we see that for every x there is only one y, and for every y there is only one x. This function is 1-to-1.
The right graph is also using the horizontal line test.
Moving the horizontal line up and down the y-axis, it is quite evident that every x is mapped onto only one y, but for every y, there is more than one x.
(i.e. x = -3, 3 y= 10 for both, and x = -2, 2 y = 5 for both). This graph is not 1-to-1
How to Find the Inverse of a Function
Why do we have to know if a function is one-to-one?
If we know the function is one-to-one, then it must have an inverse.
What is an inverse function?
The inverse of a function has all the same points as the original function...
except that the x's and y's have been switched.
The domain becomes the range, and the range becomes the domain.
We use the following notation to represent an inverse function, f-1(x).
How do we find the inverse of a 1-to-1 function?
~ Mapping
~ Set of ordered pairs
~ Graphs
~ Equations
The diagram above shows a state and its population (in millions) data.
The left diagram maps the state to its population.
Interchanging the population with the state, we now have a new mapping.
The right diagram switches the range with the domain, and we now have a mapping of the population to the state.
This was possible because the left diagram is a 1-to-1 function.
So, we can interchange the domain with the range, giving us an inverse function.
2. Set of Ordered Pairs
The following set of ordered pairs is a one-to-one function.
{(-3, -27), (-2, -8), (-1, 1), (0, 0), (1, 1), (2, 8), (3, 27)}
Domain: {-3, -2, -1, 0, 1, 2, 3}
Range: {-27, -8, -1, 0, 1, 8, 27}
Interchanging the domain with its range, we now have an inverse function of the ordered pairs.
{(-27, -3), (-8, -2), (-1, 1), (0, 0), (1, 1), (8, 2), (27, 3)}
Domain: {-27, -8, -1, 0, 1, 8, 27}
Range: {-3, -2, -1, 0, 1, 2, 3}
Domain of f(x) = Range of f-1(x)
Range of f(x) = Domain of f-1(x)
Because of this relationship we can state the following:
f-1(f(x)) = x, where x is in the domain of f(x)
f(f-1(x)) = x, where x is in the domain of f-1(x)
3. Graphing
(G) GrAssign - Inverse Plotting Points - Desmos (click on GrAssign to link to assignment)
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