One-To-One and Onto
Notes
Suppose f : A → B is a function.
A function is onto when the target set is equal to the range set.
Otherwise it is not onto.
A function is one-to-one when it passes the horizontal line test. If
every horizontal line intersect the graph of a function at most once,
then the function is one-to-one. Otherwise it is not one-to-one
because some horizontal line intersects the graph of the function at
two or more points.
Examples f : R → R where f (x) = x 2 is not onto.
f : R → [0, ∞) where f (x) = x 2 is onto.
f (x) = x 2 is not one-to-one, but g (x) = x 3 is one-to-one.
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Review: Function Composition
Notes
Ex 1 f : R → R where f (x) = x + 29 and
g : R → R where g (x) = x − 29
f ◦ g (x) = f (g (x)) = f (x − 29) = (x − 29) + 29 = x
g ◦ f (x) = g (f (x)) = g (x + 29) = (x + 29) − 29 = x
Ex 2 f : R → R where f (x) = 5x − 5 and
g : R → R where g (x) = 51 x + 1
1
1
f ◦ g (x) = f (g (x)) = f ( x + 1) = 5( x + 1) − 5 = x
5
5
1
g ◦ f (x) = g (f (x)) = g (5x − 5) = (5x − 5) + 1 = x
5
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Inverse Functions
Notes
Suppose f : A → B is a function. The purpose of an inverse function
is to ”reverse” the assignment of the original function.
The inverse function of f
A function g : B → A is called the inverse function of f if f ◦ g = id
and g ◦ f = id.
If g is the inverse function of f , then we write g as f −1 . The
ornament. It does not mean ”to the (−1)-power.”
−1
is an
A function has an inverse when it is both one-to-one and onto.
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Reversing Assignments
Notes
Examples (1) Suppose f is a function with an inverse. We are told
that f (5) = −2. What is f −1 (−2)?
f (5) = −2
5 = f −1 (−2)
(2) An object a in the domain of f is assigned f (a) = 10.
Then f −1 (10) = a
(3) Suppose g is a function with an inverse.
If g (−6) = 0, what is g −1 (0)?
(4) Suppose g (x + 1) = 4 and g −1 (4) = 27. Then
x + 1 = g −1 (4) = 27
and x = 27 − 1 = 26.
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Finding the Inverse
Notes
If you know f is an invertible function and you have an equation for
f (x), you can find f −1 .
1. Swap f (x) with y in the equation for f (x).
2. Use algebra to solve for x.
3. Swap x with f −1 (y ) in the equation you solved.
Example Find the inverse of f (x) = 2x − 1.
1. y = 2x − 1
y +1
2. x =
2
y +1
3. f −1 (y ) =
2
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More Examples - Checking
Notes
In these examples, two functions are given. By computing f ◦ g and g ◦ f ,
you can check if the two are inverses of each other.
Tip: Use order of operations, ”PEMDAS,” carefully. For instance, if
f (x) = x + 8 and g (x) = x − 8, write out
f ◦ g (x) = f (x − 8) = (x − 8) + 8 = x
Example 1 f : R → R where f (x) = x+5
3 and g : R → R where
g (x) = 3x − 5. Show that they are inverses of each other.
Example 2 f : R → R where f (x) = x−1
6 and g : R → R where
g (x) = 6x + 1. Show that they are inverses of each other.
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More Examples - Reversing assignments.
Notes
Suppose g is an invertible function.
1. If g (−2) = 1, what is g −1 (1)?
2. If g (−5) = 2, what is g −1 (2)?
3. If g (1) = 3, what is g −1 (3)?
4. Using the above assignments, solve for x if g (−x + 1) = 2.
5. Using the above assignments, solve for x if g (6x) = 1.
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More Examples - Finding Inverses
Notes
Find the equation for the inverses of each function.
1
1. f (x) = x + 3
2
2. g (x) =
x +3
2
3. h(x) = −7x +
4. f (x) =
3
4
2x
x −4
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Notes