What is a Function?
Relationship between inputs (domaindomain) and outputs (rangerange) such that each input produces only one output
Passes the vertical line test
OK for outputs to be shared
One-to-One Functions
A function is a one-to-oneone-to-one function if no outputs are shared (each y-value corresponds to only one x-value)
Formal definition: If f(x1) = f(x2), then x1 = x2
Passes the horizontal line test
a bx
y
y = f (x)
(a, f (a)) (b, f (b))
x
y
(a, f (a))
y = f (x)
a
One-to-One Functions
Horizontal Line Test
A function is one-to-one if and only if each horizontal line intersects the graph of the function in at most one point.
x
y
x
y
x
y
(a) An increasing function is always one-to-one
(b) A decreasing function is always one-to-one
(c) A one-to-one function doesn’t have to be increasing or decreasing
Increasing and Decreasing Functions
{(1, 1), (2, 4), (3, 9), (4, 16), (2, 4)}
Yes, each y-value corresponds to a unique x-value.
{(-2, 4), (-1, 1), (0, 0), (1, 1)}
Example 1:
Are the following functions one-to-one?
No, there are 2 y-values of 1, which correspond to different x-values.
This is a modified slide from the Prentice Hall Website.
Inverse Function
An Inverse Function is a new function that maps y-values (outputs) to their corresponding x-values (inputs).
Notation for an inverse function is: f -1(x) For a function to have an inverse, it must pass the
horizontal-line test (i.e., only one-to-one functions have an inverse function!)
Why do we need a new function? Sometimes we have the y-value for a function and we want to know what x-value caused that y-value.
Formal definition of an Inverse Function
Let f be a function denoted by y = f(x). The inverse of f, denoted by f -1(x), is a function such that:
(f -1 ○ f )(x) = f -1[ f (x) ] = x for each x in the domain of f, and
(f ○ f -1 )(x) = f [ f -1(x) ] = x for each x in the domain of f -1.
3 5 5Verify that ( ) and ( ) are inverses.
3
xf x g x
x x
f g x fx
( ( ))
53
53 5
35
3
x
x
53 5
335 3
3
xxx
x
15 5 3
5x 5
5x
x
Example 2:
This is a modified slide from the Prentice Hall Website.
g f x gxx
( ( ))
3 5 53 5
3xx
53 5
3
xx xx
5
3 5 3x
x x 5
5x
x
This is a modified slide from the Prentice Hall Website.
Yes, these functions are inverses.
3 5 5Verify that ( ) and ( ) are inverses.
3
xf x g x
x x
Example 2 (cont’d):
How do we find an Inverse Function?
If a function f is one-to-one, its inverse can generally be found as follows:
1. Replace f(x) with y.
2. Swap x and y and then solve for y.
3. Replace y with f -1(x).
yx
5
3
xy
5
3
xy x 3 5xy x 3 5
yxx
3 5 f xxx
1 3 5( )
This is a slide from the Prentice Hall Website.
5Find the inverse of ( ) , 3
3f x x
x
Example 3:
Swap x and y.
Solve for y.
Replace f(x) with y
What does an Inverse Function look like?
Remember, to find an inverse function we just interchanged x and y.
Geometric interpretation: graph will be symmetrical about the line y = x.
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